The free-surface bow drag of a two dimensional blunt body

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7

_{THE FREE-SURFACE BOW DRAG OF} A TWO-DIMENSIONAL BLUNT BODY

By

G. Dagan and M. P. Tulin

August 1970

This document has been approved for public

release and sale; its distribution is unlimited.

Prepared for

Office of Naval Research

Department of the Navy Contract Nonr-33k9(OO) NR

062-266

iotheek . e Onderaf deli nische Hogeschool, K'L (L1 D OC U N TA TIE TECHNICAL REPORT 117-17 sbouwkunde

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-TABLE OF CONTTS

Page

ABSTRACT

-V-INThODTJCTION 1

BRIEF ANALYSIS OF BABATS CPERIMEtTAL RESULTS (BABA, 1969) k THE JET MODEL OF THE BREAKING WAVE (OUTER EXPANSION) ₅

General

₅

The Exact Equations

6

Outer Expansion

7

The First Order Solution (General)

9

The First Order Solution for a Completely Blunt Shape

12

The Second Order Solution in the Vicinity of a

Blunt Bow

₁₅

THE JET MODEL (INNER EXPANSION) ₁₇

The Inner Expansion

₁₇

The Matching of the Outer and Inner Expansions

19

GENSOLUTIONS ₂₁

THE BOW DRAG ₂₃

SUMMARY AND CONCLUSI ONS 25

APPENDIX - THE FOURIER TRANSFORM OF THE KERNAL OF

EQUATION [21] ₂₇

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LIST OF FIGURES

Figure 1 - Baba's (1969) Experimental Result

Figure 2 - Free-surfac.e Flow Past a Semi-infinite Body Figure

3 -

Dio-dimensiona1 Flow Past a Blunt Body

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NOTATION

a - dimensionless draft of the blunt part

of the bow

b - coordinate of point B in the plane

CD D'/pT'tJ'2 - drag coefficient

- drag force; D = D'/pT'IJ'2

= cp' + 1' - complex potential; f f'g/EJ'3 (outer

potential)

1

Fr = U?/(gTt)2 - draft Froude Nt-imber

FrL = U'/(gL')2 - length FroudeNuber

g () - the strength of the pressure

distri-bution along > 0

0

fe()d_ Fourier transform of g

M=

-- the equation of the body shape;

h = h'g/U2

- the length of the blunt part of the

kr I !rr/1i12

L)

,

föIJ

-

dimensionless characteristic length of

the fine part of the bow

m - the potential of a unit pressure force

on the free-surface

em()d

-. Fourier transform of nt

- pressure; .= pt/pgJ?2

dimensionless pressure

h' (x )

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:iv

-t' - jet thickness; t = tg/ut2; E = t'/T'

- draft of the semiinfinite body

- velocity at infiiiity.

- complex velocity; w wt/U?

- the complex physical plane; z = z'g/U2; =

-

the angle with the vertical of a

straight blunt bow

'y(e) = - stretching function

52(e),..., (e), g2(),... - gauge functions in the

outer and inner expansions, respectively

= i/FrT2 = Ttg/Ut2 - small parameter

X - complex variable of the Fourier

trans form

= + i.i - complex auxiliary plane; =

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v

-ABSTRACT

The breaking wave in front of blunt bow displacement ships causes most of the wave resistance of such ships. A theoretical

analysis of the two-dimensional gravity free-surface flow past a blunt body of semi-infinite length, representing a first

ap-proximatiori of the breaking wave, is presented. The jet model has been selected for the representation of the bpw momentuni

loss.

The equation of flow are solved by a high draft Froude

number expansion. The velocity distribution, the jet thickness and the associated drag are computed by the aid of inner and

outer expansions. The results are compared with Baba's

₍₁₉₆₉₎

towing-tank tests.

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-2-In the case of a gravity flow with free-surface, the bow stagnation, also ignored by the linearized theory, is associated

with a more complex phenomenon. The pressure increase is trans-lated into the free-surface rise near the bow. At small FrT this rise is insignificant and the whole potential energy is

recovered. As FrT increases the free-surface departs from the unperturbed level and becomes steep, losses its stability and

a breaking wave appears before the bow. Energy is dissipated

locally there and the resistance is augmented correspondingly. As FrT increases further the free-surface jet starts t climb up on the bOw before returning to the water body.

Although the existence of a breaking wave at the bow of

sufficiently blunt ships may be easily recognized from

ordi-nary pictures, the problem has been studied systematically in laboratory conditions only recently by Baba (1969) in connec-tion with the development of bg tankers. A brief analysis of his results is given in the following section.

In a previous report(Dagan and Tulin,

₁₉₆₉₎

we have shown

how the nonlinear effects appear in an appropriate asymptotic

expansion for blunt ships. There we concentrate on the devel-opment of the theory for the two-dimensional case of a developed

breaking wave.

The study of the two-dimensional problem is a necessary

step in the development of a theory for typical three-dimensional

ship bows. It permits a gain in Insight at the expense.Of

rela-tively simple computations. Moreover, it may be applied in a first approximation to sufficiently flat ships as will be shown

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In our previous report we have been able to predict the instability of the free-surface in the two-dimensional case by using a small FrT consistent expansion and Taylor's stability

criterion. _{The study of the high FrT regime (corresponding}

to a developed breaking wave) is replaced here by an improved analysis.

It is worthwhile to point out the difference between the problem studied here and that of planing. In the case of a

planing plate (Squire (1957)) the solutions are based on an

expansion in which FrL is of order of unity while the slope is the smal.l parameter. The first order solutions, expressed by

a Fourier series, is poorly convergent when FrL becomes small.

In other words the conventional planing theory is valid for a

ratio between dynamic lift and buoyancy of order one. Due to the existence of the detachment condition at the trailing edge, the submersion of the planing plate cannot be assigned

before-hand.

Here we are dealing with ships moving at small _FrL such that the lift/buoyancy ratio is small and the draft is

prac-tically independent of FrL. Dynamic effects may be important near the bow, but they are localized there. For this reason

we consider here a semi-infinite body (FrL - 0) and, obviously,

disregard the trailing edge condition, but assign a-priori the

shape of th,e body beneath the unperturbed level. This way the

bow field is accurately described, while the pressure

distri-bi..ttion along the body far sternwise is eventually distçrted.

Since we are primarily interested in the bow nonlinear effects,

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-k

-BRIEF ANALYSIS OF BkBA'S EXPERIMENTAL RESULTS (BABA,

1969)

Baba' has carried put a series of towing-tank tests with three geosims of a tanker With Cb. = 0.77. The total resistance has been measured äonvent'ionally, while the wave and frictional

resistances have been measured separately by wave and wake

sur-veys, respectively.

In the ballast condition, and in a lesser manner in the full load condition, an, important breaking wave developed at the bow (Figure la). The energy loss, and the associated

re-sistance, have been ingeniously measured by a wake survey near

thebow. The experimental results from the three geosiths may

be.surnmarized as follows,'fdr the ballast conitions:

(i) The breaking wave resistance followed closely

Fróude similitude.

(ii). The breaking wave inceptiOn occurred at FrL = 0.16 or FrT 1.15.

(iii) At the maximum FrL Q.2k considered, the breaking wave resistance was 18 percent.of

the total resistance while the wave

resis-tance was of order of 6 percent. At smaller FrL the ratio between the breaking wave and wave resistance as even larger.

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-5-Baba has suggested a model of a one dimensional hydraulic jump as a crude approximation for the breaking wave, as if the flow was uniform and limited by a horizontal rigid wall at the

ship bottom elevation.

To compare his model with the experiments, Baba has assumed that the breaking wave is normal to the direction of motion (Figure ib) and has estimated its effective lateral

extent as roughly half of the beam B'/2. By dividing the measured energy 1os with this length, the two-dimensional

equivalent of the drag coefficient takes the value

C = DT/0.5pUT2TT 0.08. This value has been dbtaned in

ballast conditions for the maximum speed considered (FrL 0.2k,

FrT 1.70, Figure 1).

THE JET MODEL OF TiE BREAKING WAVE (ouTER EXPANSION) (a) General

We consider now a two-dimensional gravity flow past a semi-infinite body (Figure 2a) with a sufficiently large FrT such that a developed breaking wave exists before the bow.

In treating this problem we face two major difficulties: (1) The representation of the breaking wave, a zone of highly turbulent flow and energy dissipation, by an.ideal flow model

and (ii) the solution of a nonlinear problem by an asymptotic

expansion. As for (i) we adopt the jet model, well-known from

planing theories, su,ch that the momentum loss in the jet repre-sents that of the brealçing wave and we neglect, as usual, the

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returning flow. _{To make the problem nalytica1i treatable} we use. a high ErT expansion, i.e.. an expansion in which the

ideal free-surface rise near the stagt'iatibn _{region tends to}

infinity. . The jet model is consistent With this t-ype of ex-pans ion, but is a poor phsica1 representation of the breaking wave, since in the real case FT is not so large and the jet rises only moderately before its coilpse.

As in the treatment or other nonlinear _{problefis we can..} only hope that our asinptotic 'esults will

_{be vali& foa}

sufficiently wide range of FrT in Spite of the dealizations of the model.

(b) The Exact Equa.tions

Using a p'ocêdure followed in similar .proiem in the past

(Tulin

_1965,

_{Wu 1967),we map theflow. domain or the complex} potetitiaJ, plane f + 1i (igure 2b). The iariable are

outer variables and _{are made dimensionless by referring them} to U' and u'2/g (see notation). _{For convenience we ma f on}

the auxiliary hair plane _{(Figure 2c) by the transformation}

df t

- 1

- 71C

where t is the dimensionless jet _thickness.

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dh arg w = - arc tg

arg w arç t

where y h(x) is the profile equation. _{Th physical plane}

is mapped on the _{C plane by} dh dx

-7-The exact boundary conditions for the complex velocity

w are as follows

Re2w

t/ iw).= 0 (AJ) [2].

(si)

(sBA)

(C-.

I

1/irC

_[6]

(c) Outer Expasiori

We consider now an expansion near the state of unifopm

flow with ₌ i/Fr2 as a small parameter.

By definitionh(x) = ch1(x). We also assume that t = 0(1)

and we anticipate a later result b estimating -t _). Hence, forthe outer observer the body shrinks to the line

y = 0 and the points S,O,J col-lapse in the origin of the

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With

w= 1+5

we have at zero order

while at first order we obtain from Equations [2] -. _[7]

Re(.+

iw1 ( < 0, I.L 0 [91

w3. 0 ₍ .4

.-)

[io]

z=

fwiac

where 5] e, as a result of Equation [k].

Along the body ( > 0, = 0) Equation [ii] gives

x.=

-8-(c)w1(C)+ 52(c)w2(C) ±

z=C

h1(x) = - W]. [7-i

[8]

[12]

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Hence, with k1(C)

=

f

w1d

we fna11y obtain from the

integration of Equation [9]

(w1 + ik1)

= 0

(<Q

₌₀₎

_[13]

Im k1(C) = - h

(>o, i=o)

_[ik]

The. Equations [13] and. [1k] permit the computation of

w1,which in turn renders z.by Equation [ii].

By a similar reasoning we arrive at the following

equations for the second order:

+ 1W2) = ._Re[(w

+ 2

(

<, 0,

= 0) [151

1mw2 Irnw1 ₍

> 0,

= 0)

[16]

w2(C) -

0

_[17]

where ô2 =

2

(a)

The First Order SolutiOn (Generai)

Let us remark first that the.first order Equatioris [13]

arid

[114] are identical with those which would have be obtained,

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-10--an inner solution as well as .the higher Order terms are, however., more cotiveniently carried out in.the C plane.

To solve at first. order we adopt a procedure similar 'to that follOwed in platiing problems. (Squire 1957), i.e.

we rep1.ce the body by an unknown pressure distribution

g1(g) along 0,,. > 0 and determine if such that

qua-tioçis [13] and [1k] are satisfied.

The flow due to pressure force of unity strength acting

on the free-surface at v is represented y the ,1i,nerized

potential (Stoker 1957) ' i m(C) = e Ei[i(C-)] [18] where 'iu' Ei'(iC.)=

.f

2du

-and the integration in Equation [19] is carried out along a path entirely lying in the u lower half plane. The. singularity

of m .near = .v is of a vortex type, so that in faOt we replace

the. body by a vortex distr1:bution.

The function k1(C) has, therefore, the expressioti

k1(ç) e _[20]

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7r

fe

e

The

olition of this integral equation gives g1(),. which

turn permits the determination of k1(C) and w1(c) =

Equation [21],. with a displacement kernel, may be solved by the Wiener-Hopf technique. In fact, an almost identical equatio.n has

been

studied by Carrier et al (1966, _p.

397).

By applying the integral Fourier transform to Equation

[21] we obtain ..,. ..

M(X)G

r(X)

=

)H(X)] ...

[22]

the symbols being given in "NOTATION".

In Appendix we prove that

M(X)

= -

I

-11-k1(c)

satisfying Equation [13]. Equation [1k] becomes now

with the aid of Equation [.19]

)]g1(.,)dv

= -h1()

ir

-i

e

[e e

Ei(i\))jd

1

1

2

J2ir 1+X and since this is exactly the transform of. the kernel

con-sidered by Carrier et al (1966), we adopt at once their

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M(X) =

exp 1

-12-=7ji::x

exp f'X-.t, -1 1

r

1-u2

du 0

The separation of Equation

_[221

may be now accomplished provided that we select a given body shape', i.e. the function

h1(x).

() The First Order Solution for a Completely Blunt Shape

We consider the shape of Figure 3a, with h1(x) as follows

where' 0 < a < 1. and = 0(1). The forebody length is

assumed to be of order.c, such that in the limit process 0 the angle is kept fixed.

Under these. onditions, in the outer limit the body

degenerates at first, order in a box like shaped body, with the equation ' ' [2k] [25] [26] [27] ( .< x <

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H(X)

0

quation [22] now separates, by using Equations [2l-], [25] and

[29] into +, r + + 1 H

(x)

1

if(x)

I M (X)G1

(x)

-

M(0)

- .12u

[MX

M(0)

13 -h1(x) = -a + (1-a) (e_X -1) (x > 0) [28]

The shape of Figure _{3 has been selected for the sake of}

simplicity. Any shape with a forebody. Of order c (completely blunt) and .t = 0(1) yields the same solution in the vicinity

of the bow. When a = 0 the bluntness disappears, and the

shape is fine while for a = 1 the .aftbody is flat.

From Equation [28] we Obtain

1

N(X)

-jF

_K(X)

1-a 1 ₁₂

L 9

The application of Liouville's theorem to Equation

_[30]

requires additional information on the behaviorof Gl+(X)

near X = (i.e. of g1() near = 0). The solution of the

+ a

integral equation is unique if we assume that G1 (X) X

[30]

ixv

e

h1()dv =

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=

-Equation

_[31],

_{by inversiOn, renders g1().. The inversion}

cannot be carried out in a closed form because of the

inte-gral appearing in Equation [2k.]. We cian circumventthis

difficulty by replacing the kernel of Equation [21] by an approximate kernel (Carrier.et

_a1,.1966).

_{But this is not} necessary. here, since w are interested primarily. th the be-havior near the bow ₍ 0, X. _). The expansion of N+(X)

(Equation

[2k])

for large; X gives

and from Equation [31] 17

T

= F

[33]

111

1 a)

_x+i/J

M(X) M(0)

1

= ek2

(1 .+ +

M(X)

. .

7

x [31] [32]

for X with a> - _{(Noble, 1958).} In the present case i.t means that

g1()

for -. 0. To assume that

g1(.)

_- n -with n = 1,2,. . means to add eigenso.luti.ons to g1(). The

nature. of these elgensolutions and their elimination is dis-cussed later.

With the above restriction we get from Equations [29]

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-15-G1+(X) _{can be now inverted exactly (Noble,}

1957 p.

88),

the

result being

g1()=.±

b(/n

) ( -. ±0) [3k]

Di'eetly from the Equation [.20], or by taking into account that g1 () = -Re(w1 ± ik1),. we get immediately

w1(C)

=

o(/,c)

(cH

= C-e(ai + 2a./ [36].

Equations .

[.3]

and.

[36]

are the central reu its of our

analysis. We recognize i Equation

[35]

th,e linearized version of the jet singularity, while Eqation

_[36]

shows that at first order the free surface profile is continuous

at C = 0.

f) The Secord Order Solution in the Vicinity of _a.

Biut

ow The second order solution, satisfying Equations [15] and

[ 16], may be óbtathed in. principle by the same technique as

the first order one, but the computations become cumbersome.

If we limit ourselves

to

the near-bow solution, we may

re-write Equations

[is]

and ['16J with the aid of the first order results (Equations

[35

and [36]) in the following fOrm.

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a2

2ir +

-i6-Im w.2 = 0

Again, postponing the discussion of possible eigensolutions, the solution. ow2 satisfying Equations

[37]

and.

[38]

nèai

the bow has the following form

W2(C) = + O(n C) (c - a)

[39]

Summarizing the results obtained so far, we have

± ,+

p(fgn

c)

-. a)

[ko]

t. ca

21

t

= 1-61w1-52(w2-w12)

--± ... =

The outer solution is singular near the origin. We turn now

to an inner expansion which will provide the flow details

na.r the bow.

)

= 0, -0)

[37]

(u =

0, - ±0)

[38]

dz dC

j

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-17-ThE JET MODEL (I NNER EXPANS I ON) (a) The Inner Expansion

To represent the flow n the .vicinityof the o'igiri.

.(Figure3a) we stretch the variables as follows

( =

/y(),

= = z/y(e),

= b/y . . [11.2]

where "y(e) = o(1).

It is convenient (Wu, 1967) to expand the function

= øn (i/)

r + i.e

rather than. . itself. The Bernoulli free-surface condition (Equation [2])may be rewritten in. terms of 1- as follows .. . . . .

-2T

[.k3]

We now expand in an asymptotic series

c:1

(C) +

(e)(c) +

...

. [kk]

and by substitution in Equations [jo] _[s.] [k], [26] and

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(AJ, < 0)

(js,

0 < _<

[145] (SB,. _<

i.e. a nonlinear free-surface flOw without gravity (Figure

3c).

The conditions at infinity are provided by the matching with

the outer solution.

Only in the case of the polygonal shape of Figure 3a

the inner conditions are so simple. In a general case we have to solve an integral equation for 8 (Wu,

1957)

or to

start wIth a given e().

The solution or = - Q, Q is readily found from the

mixed problem expressed by Equation

[5]

in the form

-

1-V/r

\'+b

°T+vV

Far from the origin, for large C., has the following

expansion .

[146] -1

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= 1-2

dC

-19-Q=

1_2(_)+ 2(.J

In the same region, we obtain, for d/d

1

C

2

.7r

[2

'Jr

(b) The Matching of the Outer an Inner Expansions

Substituting

_{= C/'y}

a.nd = z:/y. in the inner sO1uton and expanding with c _{0 and ( = 0(1), we obtain from} Equa-tions [k7] and [k8].

w=

_{l_2(.V/_}

dC 'Jr 7T

[k7].

[!]

[50]

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Hence, we may summarize the result for the inner zero order.

solution expressed in outer variables, as follows

1 + ac/2r2 1

-

ae/2T2 According

-20-The outer expansion (Equations [ko] and [ki]). and the

inner expansion (Equations [k9] and [50]) match if

.11

aT2 c 2

to Equation [51] t = c2a and b = aT E

2

oir initial estimates being thus confirmed. One has also

to take noticéofthe fact that the expansion Of .

(Equa.-tion [k9]) is carried out up: to the term. of order c..

Mdi-tional terms of this order may appear from the next inner term b1(c)c1, but the analysis shows that such terms would not contribute tothe matching w.ith the Outer terms of order

2

given by Equations [1m] and.[.ki.]. .

A uniform expansion f or w . and z may be written by adding the. inner and outer solutions and subtracting their common

part. .

[51]

[52]

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Re.(_+ iw*)

Im w = 0 w _{= 0} -21-El GEN OLUTIONS

Elgensolutibns of the buter problem satisfy at any order

the equations . .

(> 0,

j = 0)

Such eigensolutioris are easily obtained by the Wiener-,.

Hopf technique from Equation [.21] with h.() = 0.. Forn

Equation [22] with H(;X) = 0 we find that the Fourier.

trans-form of the pressure distributiOn which replaces the body satisfied the équë.tion.

[531

[5k]

[55]

G*(X) 0

[s6j

M(X)

where the arbitrary polynom at the nominator stems from the

application of Liouville's theorems By the asrmptotic expan sion of M+(X) (Equation [2k]) we obtain for the behavior of

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w*(ç)

and similarly for the complex. velocity

N

d*

fi 1_j _n+3/2 n=O g -22-N

d*

3/+

[57]

where

d*

are arbitrary constants uniquely related to C. The eigensolutions of theinner problem

= T* + ie*

satisfy conditions of the type :

(<o,=o)

[59]

e*=o

(

>o,

..=C)

[6o]

and c has the following form

where n is arbitrary. satisfying the nonhomogeneous

problem, as well as the homogeneous one , has the following

expression

[58]

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-2

Cl

.C*2

exp _(c) :1 + +

[62]

Now, carrying out the matching withthe outer solution shows that C1* =0 forail n.exOepting C01. which is

un-determined. We rule out this eigensoiution

since

it in-volves either an infinite velocity in the jet at infinity,

or an infinite jet thickness., depending on whether C* is

positive or negative.

The next matchable eigensolutipn is that corresponding to A1(e) = c2 and C1 0, and it has to be eliminated on

the same ground. In fact the requirement of finitejet

velocity and thickness eliminates all possible eigensolutions

at the order we h'ave considered so far.

THE BOW DRAG

The horizontal force D acting on the bow may be found from the inner solution, by using aBlasius type of integral

Jd

=

imf

B B

=

Since is analytical in the lower half plane

(Fig-ure 3c) the integration along i may be replace by an

anti-clockwise integratior at infinity at A plus a anti-clockwise

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2k

-integration around and below J, both on half-circles.

Expending of Equation [11.6] at infinity and near the origin we obtain, with the values of Equation [5.1],

3/2

+ = 2 + +

o(c

_{) (}

-. a)

[6k]

0 W

-0

+ - =

-2 cos _)4j sin _J1 +

...

( -. 0)

[65]

Substituting in Equation

_[63]

and carrying out the

integration as explained above we get at the order con-sidered here

[66]

The same result, excepting the cos term may be obtained from the first order outer expansion by the pressure inte-gration around the free-surface singularity.

To roughly compare the result of Equation

[66]

with

Babats findings let assume that the bow is completely blunt

with = and a = 1.

For c = l/FrT2

0.3k,

we have

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-25-which is roughly four times larger than the value estimated

by Eaba. .

The discrepnc.y between the two figures may stem from

different causes: theasymptotic nature of the solution, the assumption of total bluntness adopted because of the

lack of information on the. bow shape details, and perhaps

the most important is, the thee-dimensionality of th flow,

Babats equivalent tvio-dimensional scheme (Figure la) being a rather crude apprOximation. .

SUMMARY AND CONCLUSIONS

The 'two-dimensional free-surface gravity flow past a

semi-infinite body has 'been solved by the tiethod of matched asymptotic expansiOns. 'The'l'iriearizd outer problem has' been solved by using the Wiener-Hopf technique. The solution has

been carried indetailfor a completely blunt body,i.e., a

body which at. first order degenerates Into a box-like shape profile. _{In this case the Quter solution is singular at}

first order near the bow, and the bow singularity is

repre-sented in the inner nonlinear solution by a jet. The jet

thickness and the associated bow drag have been computed. The method permits the derivation of the solution for other

shapes, like blunt round bows or fine shapes,. The bow singu-larity appears in higher order terms in these cases.

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The present approach has to be extended in some way to

the case of shi bows by taking into account the effect of

sweepthg back. This problem, as well as the treatment of different two-dimensional bow shapes, will be studed in

HYDRONAUTI CS ,

Incorporated

-27-APPENDIX

THE FOURIER TRANSFORM OF

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-28-The kernel has the following expression

=

1

_{Re .e}

_Ei(i)

[68]

where Ei(i) is defined by Equation [19].

With u = .+ is the kernel becomes

0

ir

iu

r

-s

m() =

!

Re e

J

-

du =

! ReJ

ds,

[69]

-the integration contour being transferred from -the lower half-plane to the imaginary axis of the s half-plane, leaving the pole

s = i on the right. By closing the integration contour in the first quadrant we obtain

0 and

L

-s ( -s e

ds=2iri-

/ e ds for > 0 0 0

r

-s

r

-s I e _/ e

j5jds=_J

5dsfor<O.

[70] [71]

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Hence, in both cases

1 e_S

m()

= Re e

Ei(i)

= - -

Re

/

s-ij

0

The Fourier transform of m() is given by

iX-s

M(X) = - 7TJ

J _j

Se

dds,

which by the residue theorem gives

M(X) = 1 0 -29--s-

_lx

Is e ds - 1 1

1+lxI

-s ds [72]

[73]

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REFERENCES

Baba, E., Study on Separation of Ship Resistance, Components,.

Mitsubishi Tech. Bul. No. 59, pp. 16, 1969.

Carrier, F. G., Krook, M. arid Pearson, C. E ,. Functions of a Complex Variable, McGraw-Hill, pp. k38, 1966.

Cole, J. D., Perturbation Methods in Applied Mathematics, Blaisdeli Pubi. Co., pp. 26O, 1968.

Dagan, G., arid Tulin, M. P., Bow Waves Before Blunt Ships,.

HYDRONAUTICS, Incorporated Tech Rep 1l7-l+, pp k5, 1969.

Eggers, K. W. H., On Second Order Contributions to Ship Waves

and Wave Resistance, Proc.. 6th Symp. of Naval Hrdro-dynamics, 1966.

Maruo, H., High- and Low-Asect Ratio Approximation of Planing Surfaces, Schiffstechnik, pp. 576k, Vol. 1k, No. 72,

1967. . .

Noble, B., Methods Based on the Wiener-Hbpf Technique, PergathOn Press, pp. 2k6, 1958.

Squire, H.. B., The Motiori of a Simple Wedge Along the Water

Surface, Proc. Roy. SOc., Vol. 2k3A, pp.. k8-6k, 1957.

Stoker, J. J., Water Waves, Wiley, N. Y., 1957.

Tuck, E. 0., A Systematic Asymptotic Procedure for Slender

Ships, J. Ship. Res., Vol. 8, No. 1, pp. 15-23, 1965.

Tulin, M. P., Supercavitating Flows-Small-Perturbation Theory, PrOc. of the InternatiOnal Symp. on the Application of the Theory of Functions in Continuous Mechanics, 2nd Vol., pp. ko3-k39, 1965 :

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-31-Van Dyke, M. D., Perturbation Methods in Fluid Mechanics,

Academic Press,

_{pp. 229, 1957.}

Wethblurn, G. P. Kêndrik, J.J. and Todd, M. A., Invetigátion of Wave Effects Produced by a Thin Body, DTMB Rep No

8ko,

_{pp. 19, 1952.}

Wu, T. Y. T., A SingularPerturbation Theory for Nonlinear Free-Surface Flow Problems, Int.Shipbld. Prog., Vol.

_1k,

No.

151, pp. 88-97, 1967.

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(a)

(VIEW FROM FRONT)

B2'2

BREAKING WAVE

_(b)

(UNIFORM)

FIGURE 1 - BABA'S (1969) EXPERIMENTAL RESULTS.(a)BREAKING WAVE BEFORE A TANKER; (b) BABA'S iWO-DIMENSIONAL REPRESENTATION OF THE BREAKING WAVE.

HYD RONAUTI Cs, I NCO RPO RATED

(b)

x,

z =x +

(c)

FIGURE 2 - FREE-SURFACE FLOW PAST A SEMI-INFINITE BODY: (a) THE PHYSICAL PLANE; (b) THE COMPLEX

POTENTIAL PLANE; (c) THE AUXILIARY PLANE.

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(c)

FIGURE 3 TWO-bIMENSIONAL FLOW PAST A BLUNT BODY: (a) THE PHYSICAL PLANE; (b) THE BODY BOUNDARY CONDITION IN THE OUTER APPROXIMATION; (c) THE BODY BOUNDARY

CONDITION IN THE ZERO ORDER INNER APPROXIMATION.

(b)

h(x)

a + (1 -a)(e

= I

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DDFORM 1473

(PAGE 1)

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S/N 0102.014.6700 UNCLASSIFIED_{Security Classification}

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2b.GROuP None 3. REPORT TITLE

THE FREE-SURFACE BOW DRAG OF A TWO-DIMENSIONAL BLUNT BODY

4. DESCRIPTIVE NOTES (T'pe of report an,incIusivc dates) Technical Report

s. AU THORIS) (First name, middle initial, last name)

G. Dagan and M. P. Tulin

6. REPORT DATE

August

₁₉₇₀

76. TOTAL NO. OF PAGES

kO

7b. NO. OF REFS

1k

8a. CONTRACT OR GRANT NO.

Nonr-33k9(0O) NH

062-266

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C.

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Technical Report 117-17

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th,s report)

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This document has been approved for public release and sale; its

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II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

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13. ABSTRACT

The breaking wave in front of blunt bow displacement ships causes

most of the wave resistance of such ships. A theoretical analysis of the two-dimensional gravity free-sxrface flow past a blunt body of

semi-infinite length, representing a first approximation of the break-ing wave, is presented. The jet model has been selected for the

repre-sentation of the bow momentum loss.

The equation of flow are solved by a high draft Froude number

expansion. _{The velocity distribution, the jet thickness and the}

associated drag are computed by the aid of inner and outer expansions.

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TECHNICAL REPORT 117-17

THE FREE-SURFACE BOW DRAG OF

A TWO-DIMSIONAL BLUNT BQDY

By

G. Dagan and M. P. Tulin

August ₁₉₇₀

This doument has been approved for public

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