Analyses on waves and the wave resistance due to transom stern ships

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TECHNICAL REPORT 117-11

ANALYSES ON WAVES AND THE WAVE RESISTANCE DUE TO

TRANSOM STERN SHIPS

By

B. Yim November ₁₉₆₇

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED

Reproduction in whole or in part is permitted for any purpose of the

U. S. Government

Prepared for

Office of Naval Research Department of the Navy Contract No. Nonr

_339(OO)

TABLE OF CONTENTS Fag e SUMMARY i INTRODUCTION i FORMULATION OF A PROBLEM COORDINATE DJFORMATION ₉

GREENS FORMULA FOR cp ii

ORDER ANALYSES FOR DIFFERENT TYPES OF SHIPS 16

TRANSOM STERN SHIPS

20

WAVES DUE TO A TRANSOM STERN 23

SLENDER BODY THEORY AND WAVE RESISTANCE FOR TRANSOM

STERN SHIPS 25

STERN WAVE CANCELLATION DUE TO A TRANSOM STERN

₃₀

STERN WAVE REDUCTION DUE TO PROPELLER

36

DISCUSSION AND CONCLUDING REMARKS

₃₈

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-ifli-LIST 0F FIGURES

Figure 1 - Schematic Diagram for a Transom Stern Ship and

the Coordinate System

Figure 2 - Schematic Diagram for V-Shape Transom Stern

Figure ₃

-

Schematic Diagram for the Transom Stern

Figure k - Near Field Waves Generated by a Uniform Rectangular

Pressure Distribution of Beam/Length Ratio, r = 0.1, Moving at Froude Number F =

0.67

(FL = o.k)

Figure 5

-

Transverse Crossection of the Waves Generated by a Uniform Rectangular Pressure Distribution of Beam/ Length Ratio, r = 0.1, Moving at Froude Number, F = 0.67 (FL = o.k7)

Figure 6

-

Relation Between Transom Stern Draft and Zero Elementary Stern Waves for a Ship Represented by

the Sine Source Distribution

Figure ₇

-

Relation Between the Stern Wave Resistance and Drafts of Immersed Transom Stern

NOTATION

A Cross-sectional area of a ship

B Half beam of a ship

B Half beam at the transom stern

s

d Defined in [19]

FH

V/gH

Function representing a ship hull defined by

_[23]

g Acceleration of gravity

G Green's function

h(,q)

Function representing a ship hull defined by

_[31]

H Draft at the mid-section of ship H Stern draft s k g/V2 k1 = kL L Ship length Defined in [20] P Pressure S = S5 + 55 Ship surface SF Free surface

V Uniform flow velocity at infinity

x, y, z Right handed rectangular coordinates system shown in Figure 1

Transformed coordinates through Equations [16] Perturbation potential

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SUMMARY

Waves and the wave resistance due to a ship with a transom stern are analyzed starting with an exact formulation of the general ship wave problem. Greens theorem is utilized together with the well known Greens function which satisfies the linear free surface boundary condition on the free surface. Then a

linearization is attempted for two types of ships; thin ships and flat ships. A transom stern ship is represented by the combina-tion of these two types of ships. For the analysis of transom stern ships, it is required to take at least the second order

terms in the formal development. It is found that the immersed transom stern acts to cancel stern waves. Using a limit form of Michells wave resistance formula, the wave resistance of the transom stern ship is analyzed.

INTRODUCTION

The first order ship wave theory has been formulated by Michell

₍₁₈₉₈₎

under an assumption that ships are thin or that

the beam-length ratio is much smaller than one and also that it

is much smaller than the draft-length ratio. However, since practical ships are not exactly thin but rather flat near and after the midsection, namely the draft is rather smaller than the beam, another formulation of the linearized ship wave theory has been attempted by several naval hydrodynamists in the form of slender body theory (Vossers

1962;

Maruo

1962;

Tuck

1963)

or as a flat ship theory (Hogrier,

192k).

Although the validity of these various formulations is known to depend on tne Froude number (Maruo, 1962; Yim, 1966 a) the

author felt that a more exact analysis should be carried out for practical ships.

Forthe solution in any case, the well known Greens function which satisfies the linear free surface boundary condition on the mean free surface has been used. It is also known that this

Greents function can be used for a development of the higher order wave theory incorporated with a distorted coordinate system (Yirn,

1966a). It is intended here to give a further explanation of this.

Many advantages of the transom stern design have been recog-nized for fast ships, and they would be a very good example of a practical ship for analysis. However, a prior mathematical

analy-sis of this type of ship does not exist to the best of the authors knowledge. This may be due to the difficulty not only in the

representation of the ship form but also in the analysis of the cavity like dent on the free surface behind the transom stern. We are completely ignorant analytically about the rooster tail

or the roach which looks like a cavity collapse or

anon-linear

wave breaking in the wake of the transom stern. Nevertheless, when we realize that even the two-dimensional linear analysis of the wave effect of planing ships is fruitful, the assumption of the linear free surface condition everywhere on the free surface may not: be too severe a restriction Here we assume that the water surface may be described by a sufficiently higher order differentiable sing1evalued function with respect to rectangular

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-3-In this report, it is intended to analyze waves due to a

ship with a transom stern from an exact formulation of the general ship wave problem utilizing Green's theorem with the well known Green's function. For this purpose, a distorted coordinate sys-tem (Yim, l966a) is used. Then, a linearization is attempted for

two types of ships; thin ships and flat ships. Finally, by a combination of these two types of singularity distributions, we represent a transom stern ship. The first order problem becomes simplified when the beam of an actual ship is as small as the draft, or of the same order of magnitude as the draft. However,

to analyze the transom stern, it is required to take at least the second order terms in the formal development. The most important term of second order is due to the effect of the line sink dis-tribution along the transom stern. The strength of the sink is proportional to the draft at the stern. It is shown that the

dent behind the stern is created mostly by this sink distribution. The importance of this term also shows that the immersed transom stern has an effect of cancelling stern waves when we use proper ship waterlines aft. Using the concept of elementary waves

(Havelock 1932+a), this cancelling phenomenon is studied.

Making use of a limiting form of Michell's wave resistance formula (Maruo 1962), the wave resistance of the transom stern ship is given. The optimum draft of the transom stern is studied for certain ship lines aft and stern wave resistance is minimized.

The effect of a propeller on the stern waves is also

FORMULATION OF A PROBLEM

The right handed O-xyz (or O-xiyizi) rectangular coordinate system is considered, whose origin O is located on the undisturbed

(or mean) free surface and the ship stern, with the x axis in the direction of the negative ship velocity and the z direction

verti-cally upward (See Figure 1). The flow is considered to be in-viscid, irrotational, homogeneous and infinitely deep under the free surface. Thus, there exists a perturbation potential P in the water which satisfies

= O [i]

We consider a ship with 2B as the beam; L the length; H the draft; S3 the ship surface, advancing with a steady velocity -V, pro-ducing a steady flow relative to the ship. The boundary condi-tiens for on the free surface SF, or z = F(x,y) are that the particles on the surface stay on the surface

(V- )F - F = O

[2]

x x

yy

z

and that the Bernoulli equation with zero pressure on the free

surfa ce,

;y) -

_{gx 2g'x}

+ 2 +

_y

2 +

2)

= o

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-5-The boundary condition on the ship surface _SS is that no flow into the ship exists,

-nV=O

n

or

(v-t )f + -

f2

₌ o

[k]

x x y z

where n is the outward normal into the fluid on S. We also note the physical facts that the flow at infinity is not disturbed and the waves on the free surface propagate mostly behind the ship.

To solve this boundary value problem, first we consider a Green's formula

k7T 1

(Xi, y1, Zi

)G(xi,

y1, Zi ;

x, y, z)

- (x1,y1,zi)G(xi,yi,zi;x,y,z)dS

n

[5]

where the subscript n indicates a partial derivative in the di-rection of the normal into the fluid and S = S5 + SF. G is the

Green's function harmonic everywhere in the fluid except at a point (x,y,z) = (xi,yi,z) where it has a singularity like

and

i

1/ (x-x1 )2

+ (y-yi

)2

+ (z-z1

)2

2

VG = O and G = O at infinity under the free surface. Now to make a problem easier, we consider the ship is such that B/L = e and H/L = O(e) where e is a small positive number. Then we may assume a series expansion,

(x,y,x,e) = e1(x,y,z) + e22(x,y,z) +

F(x,y,e) = eFi(x,y) + e2F2(x,y) + --,etc. [y]

By substituting this series into the Laplace equation [1] and the boundary conditions and collecting the coefficients of

(n = 1,2,---) we obtain the boundary value problems for each order of en(n = 1,2,---). The first order problem is to solve

O

with the boundary conditions

k +

=0

on S

iz lxx F

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-TI- c

L

_{+ kk}

r

r2

ir/2±ö o

-7-Let us suppose an analytic continuation to the space above the

free surface.

We also assume that

(x,y,z) can be expanded in a

Taylor series of z in any neighborhood of z = O whether the point

(x,y, o) on the z = O plane is above the free surface or beneath

the free surface.

Then the first order free surface condition

can be applied on z = O.

The Green?s function for this boundary value problem or the

potential at (x,y,z) due to a point source located at

(xi,yï,z1)

in z

O with the linear free surface boundary condition on z = O

is well known (see eg. Lunde 1952)

1 1

G(x1,y1,z; x,y,z) =

--t

sec2e e

z_zil + itu)

t-k sec2e - it sec e

dtde

-k sec2e{ z+z-i)

e

sec2esin(kwsece)de

O 'Tr/2+b 2

de

dmemw[ksec2esiri(mlz±zij

-rn cosrn z+z

ir

_{k2sec4e + m2}

where

)2

)2

r12 = (x-x1)2 + _(y-yi)2 + (z-zj

r22 = (x-x1)2

+ (y-y')2 +

(z+zi

= arc tan (Y_Yi)

x-xi

= (x-x,)

cos

e + (y-y,)

sin G [12]

= fictitious friction force which is put to zero after integration.

This G has the following nature:

G=0

in

z0

[13]

except at (x,y,z) = (x,,yi,zi) where there is a singularity like l/r with r = ((x-xi)2

+ (y-y,)2 +

(z_z,)2)

G

+kG =0

onz=0

[l-I]

xx o z

G,VG-*0 asroo inz<0

[15]

However, G on the line

(z=-zj.

y ₌ y, x > xi) is singular (see Ursell 1960; Yim 1966b). This distribution of singularities

comprises a part of the image singularities producing waves, weU explained by Havelock when he dealt with a two-dimensional doublet under the free surface (Havelock, 1926). These singularities are

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-9-allowed only outside or on the boundary of our physical dormain, or of the water. Therefore, in [io], we can easily see that the domain of points (x1,y1,zi) of G(xi,yi,zi; x,y,z) must be in

z1 O and inside of the ship. For a higher order wave theory,

the same Green's function G has proved to be useful (Yim, l966a).

However, to consider the effect of the change of a ship's wetted surface due to waves, which is important for a higher order wave theory (Wehausen, 19611), we may desire to use the Green's func-tion above z1 = O which is not allowed as long as we use our known Greens function G in [lo]. By a distortion of coordinate

(Yitm, 1966a) this situation can be resolved.

COORDINATE DEFORMATION

The higher order wave theory has been formulated using a coordinate deformation (Yim, 1966a). Here we are going to use the same deformed coordinate system in dealing with our transom stern ship. The transformation of the (x,y,z) coordinates to

the coordinates by

[16]

z = + F(x,y)

where z = F(x,y) represents the free surface, will change the expression for and its derivatives to

(x,y,z) = + F(x,y)] [17] * * =

c

F X

*

*

*

*

₂ c =

-2

F

-4

F + F XX X XX X

*

*

= - F y r1 y

*

*

*

*

2 =

-2

F - F

+P

F YY flr1 r1 Y YY Y

l

=q

*

z

=c

*

zz

Thus our problem in 'r1' space will be to solve

with the boundary condition on = 0

* *

+ò

o 1

1

=

and the corresponding boundary condition on the body surface. *

For the first order , d = p = 0, and the first order problem is

[19]

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-11-the same as that in -11-the physical space except for -11-the boundary condition on the ship surface. We now deal with the potential in the transformed space with the boundary conditions on the free surface = O and on the transformed ship surface. For

con-venience, we omit the superscript

GREEN'S FORMULA FOR

We investigate the Green's formula for in the rectangular coordinate space: ; S i-S. b F IL

- n''

G -

JJJ

GddT 2l]

If we first deal with the integral over the free surface

=

-

¡f

(PG -. G)d,dî

SF

using the free surface boundary condition for and G

-

_G)ddî

-

¡f

Gpdd

S

F

- G)d _-

¡f

Gpddî

[22]

Thus

G dS =

n

¿J

_yp

The integral over the ship surface SS can be interpreted as a distribution of sources with the source strength proportional to

which is given by the boundary condition, and a doublet dis-tribution whose strength is proportional to . In the case of

Michell's ship whose surface is represented by

- f() = o

we can write

G =fG -G +fG

n

_cc

the normal component and the surface element are respectively

/ dS =

Í,f(,c),c) (fG

- G

_{+ fGc)dd}

+ 12 + f2

dd

[25]

= - yp [f(<)G + + + ±

G] dd +

TfG

f, -

1, f

[23]

[2k]

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from [12]

Since

G(),)

=f(,)G (,e1,f,)

YÌT

- fG

=

(e1-1)f3G

(,e2,f,)

ar1d

G(,f,)

where

oe1i

e1<e2<i.

e have (See Eggers 1966, Peters

19-8)

=

r

J-)

S S yp

-

f(fr

+

:G)]dd

±

j

S S yp

-13-f(G +G)+(G -fG

_rr

)

ddq

f(G4

-

G4)d

± o

+ 2Vff2G(fl=62f)

f

fGdE

[26]

[28]

Gpd,d

_-

ffj

(;d dT

_[29]

1f we did not use the coordinate deformation, S and the curve L Syp

would have a domain above the free surface where our G can not have any meaning. It is obvious here that the lowest order term is the first term, which represents the first order potential for

a MichelUs ship. However in the case of a ship with a flat bot-tom the first integral over the ship surface SS in the Greens formula for

GdS

=

1,r1,h(,r1)) (hG

+ hG - G) dd

[30]

where the ship surface is represented by

-

h() = o

[31]

ce normai components and the surface element are respectively,

¡

h, h, - i

dS =

h2

+ h2 ± 1

ddr1 \

+h2+l

r1 J [32]

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-15-and S3 is the projection of SS = O.

By integration by parts

ffGdS

=

h(,r)(

G +

G )dd1

-

[33]

Sp Sp

where we used the relation G + G + G = O, and h = O on the

Yin

boundary of S in the deformed coordinates. The last integral in the above equation can be written as

S Sp )

+ h(,n)G

iJ]dd

k Gddq

-

¡f

h G( = eh)ad

= -

G_ --G d

Sp ¡

J

G dd

-

J J

h G(=eh)

[3k]

5Sp Sp

where e is a positive number less than 1. Thus we have, = G dS

+f f

(=h) G(o)dd

JJ

SS SSp

+ff(l-e)h3G(=eih)dd

ffh(G

+

G)dd

-

ff

Gpdd

-

fJJ

GddT [35] S Sp

For many ships, we may well consider both Equations [29] and [35] since some part of S is suitable for [23] and the other part

of S is for [31]. However, it should be noted here that neither [29] or [23] involves any derivative of f or h except in .

ORD. ANALYSES FOR DIFFERENT TYPES OF SHIPS

Micheli»s ships are considered to be thin, or B/L = o() while H/P = o(i). For a flat ship, we may consider H/L =

O(E)

while B/P = o(i). If we consider the case kL = O(i), and its

derivatives are

O(E)

in the above cases (See Yim 1966a). Thus,

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=f [

.1

VfG(=O)dd

[36]

s

s yp

For a fiat ship, formally we have an integral equation for the first order potential

=

-L

3Sp

-17-J dd

+

ff

(=h)G(=O)dd

[37]

Or by using the Bernoulli's equation at any point z = h in z O

gh-V4)

±(2+P2±2)+=O

X 2 x y z p

which is due to the fact that at infinity, the wave height is zero, V = O and the pressura P is equal to the static pressure, so that (gz ± P/p) = O, we can write

= - ±

± VP/pg)O(=O)ddî

Sp

pg

G(=O)ddî

-jr

hhG

=eh)dd

_[39]

s

s p

[38]

or =

vff

P(=h)

G(=o)dd

+ O(E2) pg

ssp

If the pressure on =h is given, this expression directly gives the first order . However, for a given shape of ship this

is still an integral equation with the boundary condition on the

ship, = - Vh

z x

In practice, ships are neither flat nor thin but rather

B/LI = O(E)

and H/L = O(E) too. Then we can claim that and its

derivatives are O(E2) except very near the ship where we know = - Vf

-= O(E). Thus, from both equations [29] and

{35]

the first order potential is expressed either by

[36]

or by

[L]

1T

Ht']

Equation

[36]

is Michells formula and [1Il] is Hogners formula

(Hogner, 192)4). We note, however, that [)4l] is different from

the case of flat ships, [)4o], and k =0(i) is one of the very important necessary conditions for its validity while

[36]

is the

same as in the case of thin ships and still holds for k

o(i)

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From [35]

47r3

If we note the fact that

-19-better in the case of lower speed, or kE = o(i), than in the

case of k = o(i), even for slender ships (Maruo i962 Yim i966a). Equations [36] and [1!i] also indicate that the lowest order term of the potential due to a ship whose draft is as small as its

beam is completely due to the second integral over SS of the right hand side of [21], which is the potential due to the source dis-tribution on the surface of ship.

It would be significant to look at one step higher order so-lution for 4, which is o(), since the other effects as are due

to viscosity or surface tension may be of still higher order then the terms to be considered. From [29], terms of O(E3) for 4 say

can be written as Sp

= G(,o,)

f2()

G [44] 2 i7Tq)3

=h)G(=O)dd

[43]

i i: / fG d

and cp also is an even function with respect to y, we can easily understand that [12] and [3] are identical within the order con-cerned, for a cruiser stern ship. Equation [12] or [3] can be evaluated after obtaining the lowest order potential, Cp2 from

[36] or [kl], and its second order derivatives Cp2 . When we

refer to [21], we note here that this second order effect, [2]

or [L3], is due to the normal doublet distribution on the ship

surface and the integrals over the free surface in Green's formula [21], not from the source distribution over the ship surface in the first term of [21].

TRANSOM STERN SHIPS

When we operate a cruiser stern ship at a certain speed or higher, the streamline inside the boundary layer near the stern separates. To prevent this phenomena, a sharply cut off stern as in Figure 3 has been considered (see Saunders 1957). It has been admitted that there are various advantages due to this type of stern, which is called the immersed transom stern. If we quote from Saunders work - "This type of stern:

Lengthens the fore-and-aft spread between the bow and stern elevation of the Bernoulli contour system.

Increases the effective length of the vessel be-cause the stern pressure disturbance appears to be abaft the actual stern.

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-21-Increases the effective length without the weight, bulk, size, wetted surface, and added friction drag of the phantom

ending.

Permits a very considerable easing of the curvature in the waterlines of the afterbody or run.

When carried to extremes, as in some motor boats, permits the greater part of the length of the vessel to be used as the entrance, with a consequent fining of the lines in the forebody and a reduction of pressure resistance on that part of

the hull.?!

However, the hydrodynamic analysis of (a) and (b) has not been worked out yet. It has been observed that complete separa-tion of the flow behind the transom needs a local Freude number at the stern FH =

V/JgH5

>

0.5

approximately, where H5 is the

draft at the stern as shown in Figure ₃ (See Saunders 1957). We consider here FM is sufficiently large so that the flow is smooth behind the transom forming a dent on the free surface. It may collapse as a cavity collapse or as a breaking wave at the end of

the dent, forming a so-called rooster tail or a roach. The phe-nomena of either a cavity collapse or a breaking wave should be dealt with separately as a fundamental problem, since even the gravity effect on the two dimensional cavity beneath a free

sur-face is unknown yet.

The condition on the dent behind the transom is from [21]

ar!d

[38]

with p = 0, the same as [9], the general free surface

surface where the gradient of the free surface may be large. Since is in general quite smaller than the maximum draft H, or the

transom stern is of the shape of V to prevent slamming, we may con-sider the free surface = O includes the dent behind the transom, Then we can now use either [29] or

f35]

for . Usually for this

type of ship it can be considered that the front part is like Michell!s ship and the rear part is like a flat ship. Thus, for

the first order, we may use

[36]

for the front and

[hl]

for the rear, and for the second order

[3]

can be used. If we integrate

[3]

by parts = 1 dEdr Sp B S O

-

I

G-d+1

k / d, k Sp

All the terms here are formally

O(E5).

However, the first and the last term are due to the effect of singularities which are

c()

distributed in -1 < < O while the second term represents the potential due to a source distribution, /k which is equal

to VH5() from

_[38]

with p = O, may be larger than

O(E2)

and is distributed on _Bs < 'ri < Bs. Namely the singularity distribution

in the second term is more concentrated than that of the other

terms. Thus the second term in the right hand side of [f5] seems

Gdr1

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23

-to be the most important higher order term. Besides the strength of the sinks these are already known, and this contributes to make the total sum of source strength to be zero so that the ship is represented as a closed finite body under the free surface. This

phenomena can be visualized if we consider a fictitious free sur-face inside the ship. Actually a source distribution ori the free surface can also be interpreted as a pressure gradient or a vortex distribution (yim, l966b). In fact, the pressure under a two-dimensional flat plate planing over the free surface is obtained by many hydrodynamists as a solution of an integral equation, where it is interesting to see an almost sudden jump of pressure near the trailing edge (See Wehausen, 1960).

Thus, it is crucial for understanding the transom stern to include the second term of the right hand side of

E5],

or the line sink distribution, in our consideration. As can be shown by two-dimensional example (Lamb, l9'l-5), the sink line produces the

cavity-like dent immediately behind the sink line. This fact for the three-dimensional case will be dIscussed in the next

sec-tion. For the time being, we neglect the effect of the other

terms in

L51.

WAVES DUE TO A TRANSOM STERN

It has been shown that a source distribution on a free sur-face can be represented by a corresponding pressure distribution

(Yim, l966b). For example, a flow due to a uniform pressure

dis-tribution on a rectangular patch of the free surface (z = O,

pg

P

o)

k

x

cos (t y-'q sin

e)

+ -k

B3

line on z = 0, x = -1, B3 y B3 and a sink line on z O,

= 1, -B3 y

B3.

If

we consider a slender ship with transom

stern whose stern draft H3 is almost the same as the maximum draft H, and the after body is almost parallel, then this ship

can be represented approximately by a source distribution for the front, and a sink distribution along the transom stern. To see the effect of the transom stern on the free surface near the stern, it may be enough to see the near field wave height due to the uni-form pressure described above. The wave height can be written

(Wehausen 1960)

ir/2

d sec

ir/2

co tdt sin(txcose)-sin(tx-lcose)

de

sec3e[cos(kx sec

e)

-

cos(k x-1 sec e)lcos (k y- sec2O sin

o)

This was computed on an IBM 1130 in HYDRONAUTICS, Incorporated by Thomas Huang and the results are shown in Figures i-l- and

5.

This figure shows clearly the cavity-like dent behind the stern, t-k sec2e

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-25-SLENDER BODY THEORY AND WAVE RESISTANCE FOR TRANSOM STERN SHIPS

It has been pointed out by Maruo (1962) that, for k = o(i),

a better approximation than Miche1ls theory can be obtained by the use of slender body theory for the wave making resistance of a ship whose draft is as small as its beam, using special care at the stern and bow. This is especially simple for calculating wave resistance within the lowest order. We have seen that the first order potential due to such a slender ship with a transom stern can be represented by a source distribution over the ship plus the line sink along the stern. If we use the wave resis-tance formula due to the source distribution (Havelock l93b)

qr/2

R =

8J

lH(8)2sec38de

[6

-Tr/2

where

H(ê)

=fm(x,y,z)exp(kzsec2e+ikxsecO+ikytanesece)dS

-

fH:(y)

_{exp (iky tan 8 sec 8)dy}

-E

s

Thus

H(812= H1 (8)H1 (e) + _(e» (8) + H + H1I

[8J

Putting

dS = dc(y,z)dx _E'l9J

we integrate H1 (8 ) by part with respect to x

i H(8)

- iksec8

f

m(0,y,z)exp(ikytan8sec8)dc

cc(o)

_fm(

-i,y,

z)exp(kzsec2 8 -iksec8+ikytan8 sec8 )dc c(-i)

_fdx ei

sec8 fm(x,y,z)exp(kzsec2e+ikytanesecß)dc

-1 0(x)

-E 5°]

where 0(x) is the contour of the cross section of Using integrations

7r/2

Ref

exp[k(z ?+z)sec2o+ik(y? -y)tan8seceseced8

-îr/2

=

e'K0

k

(z?+z)2 +

(y?

_y)2)

HYDRONAUTICS, Incorporated

-27-for small z, z1, y and y1, where K is the zeroth order modified Bessel function, y is Euler number (=

_0.5772156

. _{. )} arid

ir/2

Ref exp[ik(xl_x)secesecedO

= -Y[k(x'-xfl

_[52J

where Y is the zeroth order Bessel function of the 2nd kind, and

o

the relation from the slender body boundary condition, (Vossers 1962, Tuck 1962, Maruo 1962)

V dA(x)

M(x) =J m(x,y,z)dS -dx 0(x) E 53 J -1 -1 -1

[5kJ

where A(x) is the cross-sectional area.

Naruo obtained R1 due to H1 (e)12 (Maruo, 1962)

- _{t(y ,zl)dc}

If

_{t(y,z)log k (yIy)2 +(z +z)2 dc}

O(o) C(o)

+J

t(y,zI)dcft(y,z)logk(y!_y)2+(z!+z)2idc

c(-i) c(-i)

- A!(0)AT(_1)Y0(k) -

Ar(0)fA1(x)Y(kx)dx

o o o

where

dc = Vt dc = 2V (x,z)dz at the vertical edge

and _E55J

'iim dc = Vt dc = -V (x,y)dy at the horizontal edge

of the bow and the stern of the ship hull.

To calculate the contribution from the last three terms

of L-i-8J to [6], we first use the integral

qr/2 Re kf exp[kzsec2e + ikytanesecesec3ede ò [ kz K 2

21

= -le

o

z+yjj

k _{kz [K}

1z2+

L°2

z

Kiz2+y2]

- _{log ky for z = O and small y}

_[561

Now putting

Imjexp[ik(x_x)sece)sec2ede

_{= Wk(x'-x))}

_[57]

HYDRONAUTICS, Incorporated

-29-we cari write for the lo-29-west order wave resistance

B B

R = R1

-JH(y1)dy1JH(Y)lo

k(y1y)dy

- _{A7 (-l)A(o)w(k) - A(o)}

f

_{A77(x)W(kx)dx}

'IT IT _J

-1

assuming that H(y) is symmetric with respect to y.

Using this formula, we will be able to formulate a varia-tional problem for A(x) to minimize the wave resistance for a class of ships with transom stern as Naruo (1962) did for a class of ships with A(x) =(d/dx)A(x) = O at x = 0, -1. Never-theless, even before tackling the variational problem, we can analyze the stern wave cancellation as we do for a bulbous bow. Since the transom stern behaves as a concentrated sink whose

strength is proportional to the stern draft, negative cosine waves are produced from the stern. Therefore, the shape of the waterline to the stern should be concave as for the bow shape which needs a source type bulb (Yim l964). In fact, this fea-ture has been in general preferred for the transom stern to provide a clear separation of flow at the stern cut. A clear picture of the wave cancellation can be seen in the representa-tion of wave heights in the following secrepresenta-tion.

STERN WAVE CANCELLATION DUE TO A TRANSOM STERN We consider the ship source distribution

N

2n+l . i

h

m(x,y) = m1 (x)m2

(y) = m2 (y)

a x

-2n+l x

rì= O

in z = 0, -1< x< O, -B< y< B

nondimensionalized with respect to V and L.

Then we obtain the regular waves far aft of the ship cre-ated by

[59g

= 8kiJdyi

_{(yi )1imi}

(x1

'ir/2

X J

de sec38 cas {k1 (x-x1 )sec8d8 o

k2

Jdyi

(Yi

)f

dG cas [ki (y-yi )secG sec28)

x [sin [ki (x + C) secGS1 (-C)

+ cos (k1xsecG)S2

(o)

-

cos[ki (x+C)sece)s2 (-C)] £60]

[59

KYDRCNAUTICS, Incorporated where ç -31-N

(-i)

n m(2n) (a)

(kisece)21)

n= O fl n (2n+l) (-1) m (a) (k1 sece )2fl1

In

(n) a m(x) I m

(a)-i

n _j ' a /

x=a

m'(0)

=

i

a, k1

= kL

Thus for the odd power series of the source distribution [59J does not present any sine waves from the stern, but only cosine

waves. Besides if a2+l has alternating signs as in the power

series representation of sin x, then

_{82 (o)}

O. Thus, from the stern, only positive cosine waves propagate to far away behind

the ship. However, due to the immersed transom, we have a sink

line distribution along x = O, z = O, -B < y < B with the strength H(y)/-1-r which produces asymptotic regular waves at a large x,

B ir/2

-

LJ

dy1H (y'

)J

dB sec38 cos (k,xsecB)

T 7T

-B o

X cos [k1 (y-yi ) tane

sece

_[62]

which are negative cosine waves. Thus the wave from [621 has the cancelling effect of the stern waves in

[591,

just as the bulb waves do to the bow waves. To investigate an optimum

re-lation between h /11-Tr = rn2 (y)m1 (x) and H2 (y), we consider a

function m1 (x) = 2Trx (2Trx)3 (2Trx)5 3! +

5!

sin(2x)

_[631 in

-<x<0

2

Thus, if we denote H, the draft at x = -1/2 or at the mid-section of a ship, from

[591

and [631,

H - H km2 (y) [6+1

Substituting [631 and [611.1 in [601 and we compare

52 (o)

with

[621 to make the combined stern wave amplitude function zero for

each e. Then we obtain

H-H

s H k13 sec3O (2Tr)4 + (2Tr)e + k13sec3e 2i?k1sece + k1sece (k1 secO )3

[651

HYDR0NAUTICS, Incorporated

-33-At approximately 350 of B corresponds to the wave height on the critical line far behind the ship, while 0< 8 cot

cor-responds to transversal waves and

cot

2 < e < 900 corresponds

to divergent waves (Havelock 1957). For many different values of k, the values of (H - H)/H are shown in Figure 6. The

values of H/H at B = O are too large while those near 8 =

are too small, The absolute value of H has an upper bound for each speed such that

_V/JgH

.5 (Saunders 1957).

Havelock (1957) indicated that a large percentage of wave resistance is contributed from divergent waves which are com-posed of the elementary waves in cot < e < 900 when the sub-mergence of singularities is small and the speed is large. It

seems to be that the value of H which satisfies V/V ₌ s

which was suggested by Saunders will have a sufficient effect of cancelling stern waves for usual destroyers if we use a type

of mi (x) shown in [63 J

Now that we have understood how the immersed transom stern reduces stern waves, we may also obtain the optimum relation

be-tween H, H, k, and B

through a calculation of the stern wave resistance from [s8J . Namely the terms in [53] can be broken

down to the bow wave resistance, the stern wave resistance and the interference. Thus for the singularity distribution [63], resistance is the upper limit of the integrals with respect to x and x in

where

A(x) = kB

(1

(2x)2

(2x)4 s 2 + ) (H - H )

[67]

s

with

in3 (y)

= a constant in _[6g-i-1.

We use the integral form of Y0 and W from

[51]

and [57], integrate first with respect to x and X!, take the value at

x = x' = O, and perform the integration with respect to e, y and

y'. Also, in the integration of the first term of the right hand side of [67] in the last integral of [66], we consider B O to

evade the singular behavior. Thus, we obtain, as a stern wave

res istanc e, - o o

fdxrJATT(x)A(X)Yolkl(xT_X)ldX

- -1 -1 B B 2 ( S S +

t _H

(y)dy!J

H(y) log

ki(yt_y)jdy

¿7

_J S B -B s s o -O)

JA"

(x) W(k1x)dx -1

[66]

HYDRONAUTICS, Incorporated

-35-R 1

l6

Sl6B2(HH)2

s

15

2 4 pV2 s k1 k1 k1

20k8

2O1-8xl6

+-

+ 5

35

10

315

k1

k2B2H2

2

E2 log

S

i-3]

Iyk1

8

6r4

+ 8ir(H-H )B 2H

log

B

-

1

-

-S S S

Si

2 4 i k1 3k1

[681

This is exactly the wave resistance caused by the stern

waves in [601 and [621 .

Since this is a quadratic equation for

H

with given H, B, and k, according to the discriminant, £681

has either zeros or a minimum value larger than zero.

The

opti-mum values of H/H is computed for each F =

l/\1T,

and many

different B

are shown in Figure 6.

Although it is quite

sensi-tive with k1 , H/H is almost constant with respect to B/L for

the practical range of B/L.

The value of H/H in general is

quite large.

However since the interference of stern bow waves

needs to be considered for higher Froude numbers, the actual

optimum value of stern draft should be quite less as in the case

of the optimum bow bulb (Yim l966c).

It would be rather

de-sirable to consider the transom stern together with a bow bulb.

Anyhow, an exact wave analysis due to the stern in an inviscid

especially by the boundary layer. Nevertheless it is extremely interesting to see that the transom stern has the role of can-celling stern waves in addition to those possible benefits con-jectured by physical reasoning.

If we find a constant H and H with respect to y, H may be distributed along y such that the average value can be equal

to the constant, since in practice we may need rather a V shape transom than a flat one to evade slamming.

STERN WAVE REDUCTION DUE TO PROPELLER

It is known that the flow due to a propeller outside the slip stream can be represented approximately by a sink distribu-tion on the propeller disk according to the actuator disc theory for the propeller. If we remember that a point sink produces negative cosine regular waves, we can easily understand the regular waves produced by a propeller can have the role of de-creasing stern waves for such a stern that produces positive cosine waves as in the case of a sink distribution

_L63]

(see

Yim 1961+). If we consider the sink distribution is uniform ori

a propeller disc with the strength of sink a/V, then we can write

0 1

-

1 + (1 + CT)2

-HYDRONALITICS, Incorporated

-37-where

T/(p1J27rR2)

T = total dynamic thrust

R = radius of propeller

Now if we consider the relation between the radius of half body r, with the strength of a point source, 7TR2cY/V, in an

in-finite medium

irR2 a/V = r2 /1.

Namely, the flow due to the propeller outside the slip stream is approximately the same as that due to a half body with the radius

r at x = -.

We obtain by combination of the above two equations,

-

1 + (1 + C 2 I

T

j

This is an approximate relation between the radius of source type bulb with the thrust coefficient without considering any

pro-peller ship interaction. We know r/R can not be too large for a lightly loaded propeller. The optimum relation between the stern shape and the strength of the point sink can be easily analyzed as for the case of the transom stern. Of course the relati.ori is

a function of the Froude number and the depth of the point sink as in the case of the bulbous bow. We know the bulb size can be quite large for high Froude numbers. Although the wave produ.ced

r2

by the propeller may not be strong enough to cancel the stern waves adequately, it is quite interesting to notice that the type of transom stern represented by Equation [63 is also favorable in this respect.

DISCUSSION AND CONCLUDING REMARKS

To have our analysis reasonably well contained, we need to discuss several items which have been left out up to now.

The terms besides the line integral along the transom stern which are still formally

(3

₎ in Equation

[k5j

are made of a line integral along the side surface waterline and an area inte-gral on Ç = O inside the ship. These are essentially whole terms

of o(c ) if we deal with the usual cruiser stern ships. The

in-fluence of these terms on the regular wave due to the front part of a parabolic ship with infinite draft has been analyzed and it was found that the phase of the total wave was shifted slightly

forward due to the second order effect (Yim 1966b).

The bow waves of transom stern ships should be reduced by using a bulbous bow. The method of bulb design for a high speed

ship is more delicate than that for lower speed ships since the size of bulb should be decided considering the whole ship waves not just bow waves, otherwise the bow bulb will become tremen-dously large for certain cases (see Yim

1966c).

The problem of

evading separation of flow around the bulb is also very impor-tant (Yim 1967). Thus the incorporation of a bow bulb with the

HYDRONAUTICS, Incorporated

-39-transom stern would be a more efficient way of reducing wave

re-I sistance than any single usage of either one.

For a fast ship, as was shown before, the ship wave resis-tance largely depends upon the distribution of the cross-sectional

area. Therefore, the function of ship singularity m(x) can be

set up as an analytical function of x without specifying the de-tail of the shape of the cross section. The quality and the size of bulb depends not only on the submergence of the bulb but also on the behavior of m(x).

If we use the scheme adopted in this report, an integrated design of a fast ship with a bulbous bow and transom stern should not be too difficult. It would be very interesting to test in a towing tank a good bulbous bow destroyer with transom stern

REFERENCES

Eggers, K.W.H,, 'On Second Order Contributions to Ship Waves and Wave Resistance," Sixth Symposium on Naval Hydro-dynamics, CNR, Department of the Navy 1966.

Havelock, T. H., "The Method of Images in Some Problems of Surface Waves,"Proc. of Roy. Soc. A. Vol. 115, 1926.

Havelock, T. FI., "Wave Patterns and Wave Resistance," TINA,

Vol.

76, pp.

k3O-)4)43, l93)4a.

Havelock, T. H., uThe Calculation of Wave Resistance, Proc

Royal Soc. A, 1)4)4 pp. 51)4-21 (l934b).

Lamb, ¿ir Horace, "Hydrodynamics," Sixth Edition, Dover

Publications (19)45).

Hogner, C., "Uber die Theorie der vor einem Schifferzeugten Sellen und des Widerstandes,"Proc. First Tht. Congr. Appi. Mech. Delft, p.

146,

192)4.

Lunde, J. K., "On the Theory of Wave Resistance and Wave Pro-file»' Shipsmodelltankens meddelelse, Nr. 10. 1952.

Maruo, H., uCalculation of the Wave Resistance of Ships, the Draught of 1thich is Small

as the Beam," Soc.

of Nay. Arch. of Japan, 1962.

Michell, J. H., 'The Wave Resistance of a Ship," Philosophical Magazine

45,

_{lOb-122, 1898.}

Peters, A. S., "A New Treatment of the Ship Wave Problem," Comm. on Pure and Applied Math., Vol. 2, No. 2-3, pp. 123-148, 1948.

HYDRONAUTICS, Incorporated

-

1-Saunders, H. E., "Hydrodynamics in Ship Design," Vol. I, pp. 378-379 and Vol. II, pp. 529-531, The Soc. of Naval

Architects and Marine Engineers, New York, 1957.

Tuck, E. 0., The Steady Motion of a Slender Ship,' PhD Thesis, Cambridge, 1963,

Ursell, F., "Kelvin's Ship-Wave Pattern' J. of Fluid Mech.

Part 3, pp. 18-3l, July 1960.

Vossers, G., "Some Applications of the Slender Body Theory in

Ship Hydrodynamics," PhD Thesis, Delft, 1962.

Wehauseri, J. V., "Surface Waves," Handbuch der Physik 9, Springer-Verlag, 1960.

Wehausen, J. V., "An Approach to Thin-Ship Theory," Proc. of International Seminar on Theoretical Wave Resistance," Vol. Ii, Univ. of Michigan, l96.

Yim, B., "Higher Order Wave Theory of Slender Ships," KYDRO-NAUTICS, Incorporated Technical Report 503-1, l966a.

Yim, B., "Singularities ori the Free Surface and Higher Order Wave Height Far Behind a Parabolic Ship," HYDRONAUTICS, incorporated Technical Report 503-2, 1966b.

Yim, B., "Analyses ori Bow Waves and stern Waves and Some Small-Wave Ship Singularity Systems," Sixth Symposium on. Naval Hydrodynamics, ONR,Department of the Navy, 1966c.

Yim, B., "Some Recent Developments in Theory of Bulbous Ships," Fifth Symposium on Naval Hydrodynamics, CNR, Department of the Navy, l964.

Yim, B., "A Ship with a Cylindrical Bulb Horizontally Oriented at the Bow," HYDRONAUTICS, Incorporated Technical Report

FIGURE 1

- SCHEMATiC

HYDRONAUTICS, INCORPORATED

z

-H

FIGURE 2 - SCHEMATIC DIAGRAM FOR \/-SHAPE TRANSOM STERN

z

FIGURE 3 - SCHEMATIC DIAGRAM FOR THE TRANSOM STERN

FIGURE 4 - NEAR FIELD WAVES GENERATED BY A UNIFORM RECTANGULAR PRESSURE DISTRIBUTION OF BEAM/LENGTH RATIO, r MOVING AT FROUDE NUMBER F=0.67 (FL= 0.47)

HYDRONAUTICS, INCORPORATED 1.6 1 .2 0.8 0.4 -0.4 -0.8 -1.2 -1.6 I I 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

FIGURE 5 - TRANSVERSE CROSSECTION OF THE WAVES GENERATED BY A UNIFORM RECTANGULAR PRESSURE DISTRIBUTION OF BEAM/LENGTH RATiO

rO.1

/

MOVING AT FROUDE NUMBER, F = 0.67 (FLO.47)

Pg7

o

o

0.2

0.4

0.8

1 .0

FIGURE 6 - RELATION BETWEEN TRANSOM STERN DRAFT AND ZERO ELEMENTARY

STERN WAVES FOR A SHIP REPRESENTED BY THE SINE SOURCE DISTRIBUTION

o 0.4 0.8 1 .2 1 .6 RADIANS - 9 1.0 0.8 0.6 0.4 0.2 o =

I

0.6

HYDRONAUTICS, INCORPORATED 7 400 1 200 1 000 800 600 400 200 o -200 o F = 0.50

/

0.2 0.4 0.6

H/H

H = DRAFT OF IMMERSED TRANSOM STERN

H = DRAFT OF MIDSHIP

2B = BEAM

0.8 1.0 1.2

FIGURE 7 - RELATION BETWEEN THE STERN WAVE RESiSTANCE AND DRAFTS OF IMMERSED TRANSOM STERN

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UNCLASSIFIED

Security Classification

D D

I J&NFORM4 UNCLASSIFIED

Security Classification

DOCUMENT CONTROL DATA - R&D

(Security cla.aifcatir.i of title, body of abstract and indexing annotat,on must be entered vhen Ihe overall report ta classified)

t. ORIGINATIN G ACTIVITY (Corporate author)

HYDRONAUTICS, Incorporated, Pindell School Road, Howard County, Laurel, Maryland

2 a REPORT SECURITY C LASSIFICA TION

UNCLASSIFIED

2b GROUP

3. REPORT TITLE

ANALYSES ON WAVES AND THE WAVE RESISTANCE DUE TO TRANSOM STERN SHIPS

4. DESCRIPTIVE NOTES (Type of report and Inclusive dates)

Technical Report

5. AUTHOR(S) (Last naine, first name, initial)

Yim, B.

6. REPORT DATE

November 1967

7e. TOTAL NO. OF PAGES

51

7b. NO. OF REFS

21

8a. CONTRACTOR GRANT NO.

Nonr

339(OO)

b. PROJECT NO.

d.

95. ORIGINATOR'S REPORT NUMBER(S)

T.R.

117-11

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10- AVA IL ABILITY/LIMITATION NOTICES

Distribution of this document is unlimited

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Office of Naval Research Department of the Navy

13. ABSTRACT

Waves and the wave resistance due to a ship with a transom

stern are analyzed starting with an exact formulation of the general

ship wave problem. Greens theorem is utilized together with the

well known Green's function which satisfies the linear free surface

boundary condition on the free surface. Then a linearization is

at-tempted for two types of ships; thin ships and flat ships. A

tran-som stern ship is represented by the combination of these two types

of ships. For the analysis of transom stern ships, it is required

to take at least the second order terms in the formal development.

- It is found that the immersed transom stern acts to cancel stern

waves. Using a limited form of Michell's wave resistance formula,

Wave resistance, Transom stern

Higher order wave theory Froude number

Bulbous bow

Greerits funetion

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