Bow waves before blunt ships

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BOW WAVES BEFORE BLUNT SHIPS By

G. Dagan and M. P. .Tulin

December

1969

TECIUlICAL REPORT 117-14

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-3349(OO)

TABLE OF CONTENTS

Page

I. INTRODUCTION... 1

II. INNER AND OUTER EXPANSIONS AND CLASSIFICATION OF

SHIPS... 2

Notation and Basic quations 2

Outer Expansion; Classification of Hull Shapes k

Inner Expansion; Bow Singularity...7. III. GRAVITY FLOW PAST TWO-DIMENSIONAL BLUNT BODIES

OF SEMI-INFINITE LENGTH .. 13

I. General . . .

. 13

Free-Surface Gravity F10 near a Stagnation.

Point . 15

SmaliFroude Number Flow (FrT<l) 17

k. High Froude NUmber Flow (FrT > 1): The Jet Model 26

IV. CONCLUSIONS .

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LIST OF FIGURES

Figure 1 - Notation for Steady Flow Past Ships

Figure 2 - Ship Data Relating Beam/raft and Froude Numbers Figure

3 -

Two-Dimensional Flow Past a Body of Infinite Length Figure 4 - Free-Surface Flow in the Vicinity of a Stagnation

Point

Figure 5 - Flow Past a Blunt Body at Small Froude Number

Figure 6

-

Flow Past a Rectangular Body at Small Froude Number Figure 7 - The Free Surface Shape in Front of a Rectangular

Body

Figure 8

-

Minimum Pressure Gradient versus Fr.

NOTATION

Dotted va'iabIes are dimensional; undotted variables are.

dimensiOnless .. . .. ,,.. .

a, b constants

31 . ship beam

drag force

f, F . outer and inner complex potentials.

FrT Ut2/gT?

Froude numbers FrL = U'2/gL'

g gravity acceleratibn

h'(,z'), h?(xt) functions describing the hull hape

k(z) function of complexyárable (k =w + if.)

L! . shiplength .

forebody length

free surface elevation (inner., dimensiônless

p1 . pressure .

P . pressure (inner, dimensionless)

q velocity modulus .

T1 draft

u', v' velocity components .

/' _w1 velocity component (Sect.. II), complex

velocity wt

= U' -

iv' (Sect. iii)

-U' unperturbed velocity at infinity

U, V, W velocity components (inner dimensionless Sect. 11); w = u - iv (Sect. iii)

V velocity vector

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-iv-Zr coordinate (Sect. ii), complex variable z' = x' + iyt (Sect. III)

X, Y, Z coordinates (inner, dimensionless)

constant jet thickness (t = E = TVLt draft/length ratio = B'/L beam/length ratio = T'/'gU'2 velocity potential

velocity potential (inner dimensionless)

CD, auxiliary complex variables

auxiliary variables

angle and also dummy variable

e angle between velocity vector and x axis

I.

INTRoDUCTIoN

The wave pattern created by a ship mbving steadily in an

ideal' fluid and the related. wave resistance are classical subjects

of hydrodynarniOs.

Although the theory has diversified and

corn-putational refinements have been achieve

with time, there has

been little essential progress beyond th,e linearized techniqUes

introduced by Michell and Havelock.

In essence the present method

of solution of the gravity flow problefn is based on two

approxi-mations:

(i)

the free surface condition is linearized and' '(ii)

the hull is replaced by a singularity distribution along a line

or.a plane.

The wave resistance is generally determined frornthe

rate of energy radiated far away from the ship..

The above two basic approximations have beet given foundation'

ma rational way in the last'years by the alication'of the

method of matched asymptotic expansions (Tuck 1965, Ogilvie 1967).

It has been shown that the classical theory is in fact' a first

order term of an outer expansion in which the observer is fixed

with respect to the ship length while the ratio draft/length or

beam/length (or both) tend to zero and the Fr number based on

length remains constant.

In the vicinity of the body, in the

inner zone, the solution is still valid, provided that the

slen-derness' parameter is sufficiently small and the ship has a fine

form.

The latter condition has been somehow overloOked 'when

applying the theory to actual ships.which do not generally 'have

a needle-like or knife-like shape.

In the extreme. case of a

blunt s'hape th.ere is stagnation at the bow and the linearized

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-2-assumptions are badly violated there: the speed and the free surface rise are no longer small perturbations of the uniform speed and of the horizontal level, respectively. At the stern the situation is different due to separation and viscous effects.

Realizing the importance of bluntness effects on resistance of real ships, we have initiated a study of the free surface flow near the bow and of the related resistance.

The present report summarizes our first results which, be-cause of the complexity of the problem, involve in this initial stage rather crude approximations obtained for highly schematized

configurations. It is our feeling, however, that this initial step yields basic understanding of the problem. We hope to be able to extend and refine the results in the future, to compare

them with experiments, and eventually to apply them to actual

ships.

II. INNER AND OUTER EXPANSIONS AND CLASSIFICATION OF SHIPS

1. Notation and Basic Equations

The symbols used In this report are given in "Notations't and also shown in part in Figure 1.

The ship Is defined by the shape of its hull represented by the equation

f(x', y', [2.1]

The three basic lerigths associated with the hull are B', Tt

and L'. Additional geometrical coefficients or engths may be

consider?d, like the fo'ebody length L1'. The latter is irn-portaht in characterizing the bluntness.

Assuming that the flow is steady and uniform at infinity, the equations satisifed-by V' and.i', given here for convenience of reference, are as follows:

rot V = 0 [2. 3

in the flow domain) dlv ' = 0 [2.k] u'2+

v'2+ w'2

2

+gi=

Ufl', -

win!,

= h!(xi, z') _[2.2]

I (on the free-surface

= y1t

(X1,Zt))

-

u'ht , - w'h' , = 0

XI z [2.7]

Equations [2.3] and [2.k] express as usual irrotationality and incompressibility, Equation [2.5] is. the dynamic Bernoulli coildition on the free surface, -while Equations [2.6] and [2.7] are the kinematical boundary conditions along the free-surface and the hull respectively.

[2.5]

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-k-In order to render the solution unique, the radiation con-dition is imposed

U' = - U'; v' = w' = 0 (x' + oo;

y'. - o) [2.8]

Equations [2.3] to [2.8] may be reformulated in terms of the velocity potential ' by replacing V by grad '.

2. Outer Expansion; Classification of Hull Shapes

Economy is achieved by making the variables and the ëqua-tions dimensionless in the standard way. Let us define the following outer variables

V=v'/tJ'(u,v,w=u'/tJ',v'/tJ',w'/tJ'); x,y,z =x'/L',y'/L',z'/L',

= r'/L'; h = h'/L'; p =

pt/pu'2;iP

The equations (v)2 of flow [2.3] - [2.8] = 0 (in the rl/FrL2 = 1/2 [2.9] become, in terms of P, flow domain) [2.10] [2.11] + 2 = -

_'''

- rl, = [2.12] ,y -u = -1; x -x v = w , z z 4, h, z z = 0 = 0 (y = h(x,z)) (x + co; y -[2.13] [2.l1]

The solution of depends on x, y, z and the pärarnêter.s

FrL =Ut/(gLt)2, B'/L', T'/L for a given hull shape. The clas-sical technique for simplifying the nonlinear problem is to take advantage of the fact tht /L' or T/L or both are much smaller

than unity. With = T?/L! arid _{= Bt/L! an o.ter expansion is}

obtained by assuming that V and may be exp±'essed as a series assoiated with an asrmptotic Seuence based on

or . This

has been done in numerous publications (see for instance Wehausen and Laitone,

₁₉₆₀

and Tucks

1965)

and will not be repeated here. Since by definition

h (x,z)= (x,z)

_[2.15]

where H =. 0(1), it is natural to consider anexp.ansioh of the type

= - x +

+ O()

Ti +

()

[2.16]

with x, y,z = 0(1). Foi- an outer observer with a position filxed tith respect to the ship length, the ship collapses in a line or a plane at zero order and the flow is unperturbed. At first

order (and we consider here only first order terms) the equations become the well known linearized equations

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The hull, at first order, degenerates into:

( i) a line in the case of slender ships = 0(E)),

a Vertical plane in the case of thin ships ( = 0(1) and = 0(1)) and

a horizontal plane at y = 0 for flat ships = 0(1) and E = 0(1)).

Different flow regimes, and equations accordingly, are ob-tained corresponding to the relationship between FrL and E. OgilVie (1967) has analyzed these possibilities. Since we

con-centrate here on displacement ships mainly, we should consider the following two possibilities:

(i) Small FrL number, FrL2 = 0(1). In this case a

direct expansion of Equations [2.10] - [2.11] gives

-6-V2i

= 0 y < 0 [2.17]

-Ui +

1-ii/FrL2 = 0 [2.18] (y = o) Vi

_-

fli, = 0 [2.19]

v1+H, =0

x (on the hull) [2.20]

U = Vi =

wi = 0

(x+oo;y. -

cG) _[2.21]

i.e., a rigid wall conditionof the free surface. This-expansion is discussed ±ndetail.iri Section.III.3.

(n)

FrL = o(i), which yields the ordinary ship ré-sistance prOblem, with gravity waves left behind

the ship (Eqüatibts [2.17] - [2.21.].)

Higher FrL lead to planing problems not cnsidered here. Problem (ii), by far the most interesting, has been solVed by replacing the degenerated hull by: (i) a line of sources for slender ships, (ii) a source distribution in the mid-plane for thin ships and (iii) a .pressure distribution on. the free surface for flat shlps (Lunde, 1952). .

The fulfillment of the free-surface conditiohs [2.18] and [.2.19] is equivalent to th extension of the flow in the whole space (above y = a) and the introduction of an ininite system of singularities in y > 0, reflectioti of the ship singularities. In the case of small FrL just one i.niageissufficientinordert satisfy Equation [2.22]. . . . ,. . . .

3. Inner Expansion; Bow Singilarity .

The outer expansion is singular near the body in the FrL.= 0(1) case sinbe the first prder velocity tends thOre to infinity. Tuck (1965) has considered n inner expansion for slehder ships. The inner variables are defined as

X = x = x'/L'; Y = y/ = yt/T!; Z =z/ =

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while X,Y,Z =

o(i).

In the inner limit the observer is fixed with respect to the beam (or draft). For such an observer, when 0 the hull cross section keeps its shape unchanged while the shiplength tends to

infinity. The equations become, at zero order, two-dimensional

(in y,z) and the free surface condition becomes that of a rigid

wall.

The matching of the outer and inner expansions yields (Tuck,

1965)

the classical result, at first order : i.e. the replacement

of the ship by a source system. The source strength is propor-tional to the cross-section area variation. The inner expansion is valid only if this variation Is gradual, i.e. for fine ships. The slender body expansion fails in the bow region if the ship has some bluntness, and there the inner problem Is no more one of two-dimensional flow in the Y,Z plane, nor Is the, condition on the free surface one of a rigid wall. For this reason we should call the slender body expansions outer and inner rnidbody expansions, in order to stress their limitations.

-8-The equations of flow[2.lO] and [2.11] are again expanded by assuming that U,V,W and N are asymptotic series with respect

to

u,v,w

= U

,v ,w

+ o()

0 0 0

N = N

+ o()

In the case of a blunt-bow ship the appropriate inner vari-ables in the bow region are

X,Y,Z = x/,y/,z/; U,V,W = u,v,w; N = ₌

[2.25.]

U,V,W = U,V,W

+ o();

=

i(x,y,z) + o();N

= N+

o()

2.26]

with X,Y,Z = 0(i).

While in the case of the inner midbody expansion the ob-server is fixed laterally with respect to the ship and at zero order the length tends to infinity, in both bow and stern direc-tions, in the bow inner expansion the observer is fixed with re-spect to the bow and the shiplength tends to infinity sternwise.

Substituting [2.25] and [2.26] into Equations [2.10] -[2.13] we get at zero order

0 = 0 (in the flow domain) [2.27]

U2

+V2

0 0 0 + N 2

20

Fr

V -UN

-WN

=0

0

oo,x

Oo,z

T

V -UH, -WH, =0

0

0 .X

0 Z ( (z=N ) 0 [2.28] [2.29] = H) _[2.30]

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The conditionat infinity [2.1)4] is lost and is 'epIaëd by the matching with the outer expansion..

The bow Froude number FrT= U.1/(gTl)

_{is related toFL ad}

through .

i.e. free gravity flow at zero order in the inner region.

(ii) _{FrL2 O()(l/FrT2 =0(1)).} Equation [2.28]rernains unchanged and we have the full nonlinear gravity problem.

(ifl) FrL2 _{= O(2)(l/FrT2}

=o(i/)).

This case reduces to

that of a rigid wall condition' discussed in the

p.re-cedIngec.tion. .

In the case of. slender and tiin ships the flow in the

vicinity .of the bow is three-dimensional in all the above

approxi-matiohs. Further simplifications are achieved in the case, of

flat ships. Then he proper inner váriablës are

X = x/; Y = y/; Z = z.; N = r/; c = P/; U,.V,W = u,v,w . [2.33]

u 2 + 2 ± W:2 =1

0 O 0

=N.)

0

2.31]

32] Consequently the bernoulli Equation [2.28] may have thefollowing form depehding on the order. of magnitude of Fr:

while the asymptotic expansion starts as

=

+ 0(); N

N + o(); Y,V = TJ,V + 0(E); W = o()

[2.3k]

Again the substitution of [2.33]

and

[2.3k] into Equations 12.10] - [2.1k] gives, at zero order

2

=0

x,y 0

u2 +v2

_{0 0} +

12N

02

-2 FrT

=N

0

V -UN

=0

0

00,X

V -UH, =0

(Y=H)

0 0 X [2.35] [2.36] [2.37] [2.38]

and the problem is reduced to that of gravity flow in the ver-tical X,Y plane in the vicinity of a body of shape Y = H(X,Z)

(here Z appears as a parameter). The requirement of flatness has the meaning of T'/B' < 1, but still allows for B?/L1 < 1. In the flat ship approximation the observer attached to the bow sees both width and length tending to infinity (although pos-sibly at different rates). Obviously this approximation is nat valid near corners or regions of large change of H with X. There the full three-dimensional flow or some other approximations have to be considered. Again we obtain the three differthteases

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-12-We summarize the discussion of all the encountered cases in the following table:

DISPLACEMENT SHIPS (FrL <

obs.: In the ooe of fi.e ships the midbody expansion is valid everywhere.

FrL2=UT2/'

0(1) 0(i)

c()

Fr=u/gT

0(E)

o(i)

f_i 1 )

Outer Expansion Rigid A Wail I 'I _{Rigid Wall}

Linearized gravity waves far from the ship. The ship is replaced by a line (slender)

or a plane (thin and flat)

distribution of singularities. Condition Condition Inner t Midbody Expansion Everywhere Rigid Wall

Rigid Wall Condition The flow the of perturbed un-is state rest. I

lender ships: two-dim. flow n vertical planes normal to the centerline (Tuck, 1965). Condition

Inner

Bow

Expansion

Nonlinear Gravity Nonlinear free-gravity flow Flow. Slender and

Slender ships: three-dim.flow. Thin ships: three-dim. flow

near a strut. Flat ship: two-dim. flow in

vertical planes normal to a body of infinite length. thin

Tp:

three-dim. flcw flat ships: twc-dim. flow in vertical planes normal to the, bow.

Finally we present in Figure 2 a plot of BT/T! and FrT for more than one hundred existing ships.

There is no apparent correlation between the two parameters. At any rate most of the ships considered are flat rather than

thin (B'/T' = 2.2

3)

The draft Froude number FrT is of order one inmost cases, but reaches values as high as 2 for a rapid containership and more than 3 for cruisers and destroyers.

III. GRAVITY FLOW PAST TWO-DIMENSIONAL BLUNT BODIES OF SEMI-INFINITE LENGTH

1. General

In the preceding sections it was shown that in the case of flat ships the inner bow flow reduces to a two-dimensional flow in a vertical plane normal to the bow. In the remaining sec-tions of this work we consider exclusively such flows. More-over, we are assuming that the outer flow is also two-dimensional and that the body is of semi-infinite length. Obviously, these assumptions simplify the problem considerably. The essential features of the bow flow are, nevertheless, included in the

picture. We plan to apply the results by some approximate tech-niques to actual ships in the future, taking advantage of the fact that for most ships the ratio draft/beam is smaller than

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-1k-Consistent with the range of FrL considered, which apply to displacement ships, we assume that the bottom of the midbody is horizontal. The results permit however, to compute trim to a first approximation, but we do not consider this problem here.

In the case of a two-dimensional flow (Figure 3) the dimen-sionless velocity potential depends on only one parameter for a given hull shape: = _(x,y;FrT). Consequently, the possible asymptotic expansions of the exact Equations [2.10] - [2.1k],

with the z components deleted, reduce to the following cases:

Small FrT . In this case the Bernoulli equation

gives a rigid wall condition in a first approximation. A uni-form expansion solves the problem. Results for the first and second order approximations are given in Section 111.3.

Large FrT. In this case the outer flow conforms to equations similar to the linearized Equations [2.17]-[2.2l], while the inner flow is that of a free-gravity flow at zero

order. For this regime we suggest two possible inner flow

models: The jet model discussed in detail in Section III.k and

the spiral vortex model. It is presumed that the jet model is adequate for large FrT, while the spiral vortex model represents moderate to large FrT flows. Only the theory for the former is presented herein.

In addition a discussion of the exact equations of free-surface gravity flow near a stagnation point is presented in Section 111.2.

2. Free-Surface Gravity Flow near a Stagnation Point

Let us consider the confluence between a free-surface and a rigid wall in the vicinity of a stagnation point (Figure ka).

In the symmetrical case (?'i = -

?2)

the classical Stokes

result (Wehausen and Laitone,

1960)

requires that ? =

-= 120°. This result will now be extended for other possible angles between AO and OB.

In the vicinity of 0(z = 0, Figure 5a) we assume that the z-plane is mapped on the complex potential plane f (Figure 5b) by

-i7j ?/ir

z = a e f + R(f)

13.1]

AOB being obviously a stream line.

The function R, which has to vanish at 0, is assumed to be in the vicinity of 0 of the form

R = b [3.2]

with b = bTe1 a complex number and y a real number.

Obviously, y > 7\/ir, otherwise the mapping of the corner AOB is not ensured.

In order to apply the Bernoulli equation along AO let us determine y and q2 = u2 + v2 as functions of . Erom

Equa-tions [3.1] and [3.2] we obtain on AO (f =

HYDRONAIJTICS, Incorporated q2 =

H2

_2(X/ir_I)[

-i6-=

x2

_{+ y,}

(2(/)

+ 2

ab'y

By expanding Equation [3.k], q2 is found as

-

2

by

cos(+)

+....]

[3.5]

a?

Substituting y and q2 into Bernou1IiT equation and retaining terms

(2-?/) 2

of order or c at most we get

y + y .

2 -2(x/7r-1)

= - a Sin 7\1P + b' sin +

-

(i-)

+..

[3.6]

The identity [3.6] yields the following relationships

be-tween 7\i and 7:

7.

0. The first two terms of [3.6] give

= 21T/3

[3.7]

- 8

a3

= g sin 7\1

This is Stokes classical result. Obviously, 7\2 > 2IT/3 7 = 0. The first term of Equation [3.6] vanishes, and the remaining give

Since y > ?\/7r Equation [3.8] shows that ? < 2ir/3. A pa.iticular

case is that of y = 1, which renders the function R analytical In this case Equation[3.8] gives'?. ir/2,i.e. the. confluence between a horizontal free-surface and a vertical.wall.

In conclusion there are two possible angles between a free-surface and a rigid wall at the stagnation oInt:

(,i) if the wall is inclined with respect to the.

hori-0

zontal at an angle larger than 120 (2/3 < ? < ir) s. . the free surface intei'sect the wall at 120°

( = 2ir/3) and,

0

(ii)

if the wall is inclined at less than 120

(? < 27T/3) the free-surface is horizontal

= o). . .. . . . .. .. ..

We will considerblunt bows. of,the iatter type in

Sec-tionII.3.

5,.. . . .

3. Small Froude Number Flow _'T

<l)

(a) General

We consider here the flow pa.st a blunt body of the shape of Figure 5a. An asymptotic expansion with FrT as a small parameter has as its zero order term the state of res.t

cor-responding to FrT 0. Hence., it is appropriate to make the

variables dimensionless in the following way

X = xt/T?;Y = y'/T';N = '/T';H =.h'/T';U = ut/(gT?)l/2

V vf/(gTt)V2;F + iW = ('+ ?)/g1/2T?3/2;P pt/pgT2

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-18-The exact boundary conditions of two-dimensional flow are:

u2+v2NT2

[3.10]

(Y=N(x)

V_UN,x=O

_J

[3.11]

V_UH,x=O

(Y=H(x))

_[3.12]

V = 0,N

= 0,U = -

_FrT

_{(IZI *.o) [3.13]}

W =

U - IV

and F = + i' being analytical functions of

Z =

X +

iY.

(b) Small Perturbation Expansion

In order to simplify the nonlinear problem we seek a solution valid for small

_FrT.

_{When FrT}

0, while X,Y = 0(1)

the flow tends to rest (Equation 3.13) while the body retains its shape, the thickness being equal to unity.

It is a matter of simple algebra to show that a nontrivial small perturbation expansion has the form

U ₌

_{FrT U1 + FrT3U2 +}

N ₌

FrT2N1 + FrT4N2 +

The above expansion will be shown to be regular at infinity and consequently there is no need to consider inner and outer

expansions separately.

Expanding

u(x,Y)

and V(X,Y) in the vicinity of Y = N(X) as given by Equations [3.14] and substituting in Equations

[3.10]-[3.13],

we get the following set of equations after separating

Hence the first order approximation is that of a rigid wall on the free surface and uniform flow at infinity.

(ii) U2, V2, N2

V2 _{= (U1N1),x}

(x > a, y

= a)

_[3.19]

N2 U1U2

(x > a)

[3.20]

V2

_-

152H,X = 0 (Y = H(X))

_[3.21]

terms of the same order (i)

u1, v1,

N1

V1 = 0

(x > 0,

y

= 0)

_[3.15]

Nj. = . (i -

U12)

(x

> 0, Y =

o)

[3.16]

Vi _{- U1H,x} = 0 (y .H(X)

[3.17]

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V = 0, U2 = 0

2

-20-( _I -# oo) [3.22]

In the second order approximation, the condition on the un-perturbed free surface (Equation 3.19) is equivalent to a dis-tribution of sources generated by the first order flow, with no flow at infinity. It is easy to ascertain that

fv2dx=NlVl

=0

[3.23]

Since N1 0 as X -. co _{and U1 = 0 at the stagnation point X = 0.} Obviously W1 and W2 are analytical functions of Z.

Higher order terms satisfy equations similar to those of second order, but the computations become tedious as the order is increased.

c. General Solution

The solutions of the different order approximations may be obtained as follows (Figure

5):

The region AOBA. of the Z plane is mapped on the half plane by

z = z()

[3.2)4]

and the first order complex potential F1 =

i +

iW1 is mapped on the same i plane by

At second (and higher) order the imaginary part of

F2=+ i'2

is given along the ' axis (Figure 5b) by Equations [3.19] and

[3.21]

= -U1N1 ( > 1)

[3.26]

= const ( < 1)

and F2() is found by solving the related Dirich1t problem.

(d) Application to the Rectangular Body

As a simple example we consider the box-like shape body of Figure 6. The AOBO region of the Z plane is mapped on

the half-plane by

Z 1

(2

1 +

(2

_i)]

_[3.27]

where both

(2_ 1)2

and the logarithm have real determination on

> 1.

With F1 = - we get

W1 =Ui - 1V1

=-_

-

(u-)

[3. 28]

In particular from Equation [3.16] N1 is given by

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-22-Equations [3.29] and

_[3.27].

describe theshape of the free surface in a parametric form. N1(X) is represented graphically in Figure 7.

The next order term F2 = + _1?2 has, according to Equa-tions [3.21], [3.28] and

[3.29],

the imaginary part

= - _U1N1 -

(>

1, = 0) [3.30]

'i12 = 0 and W2 vanishes at infinity.

F2(), with given imaginary part on the real axis is determined by the Cauchy integral

F2(C) =,

-f;2c

-

(:+')

[3.3'] 1

The integration in Equation [3.31] may be carried out analytically, the result being for = , , > 1

1

1 2

(\2

£n[(2

₁₎

=

1T

k')

+ 1

The velocity

u2(o,)

is, accordingly

u2(u1 =

= 3 :

)+

[(2

₎₍

[3.33]

and finally the free surface elevation is given by Equation [3.201

1

N2 = - U1U2 =

_{3 +}

1

[(2

.l)i)

[331]

(- l)

J

N2 and N/FrT2 = N1 + FrT2N2 as funtions of X are represented graphically in Figure 7.

(e) Pressure Distribution and Forces Acting on the Body

The dimensionless pressure has the following expan-sion resulting from the Bernoulli Equation,

2

= - [l-(Ui ± V12]- FrT4(U1U2 +

V1V2+.

[3.351

A detailed analysis f the forces acting on the body show that the drag is equal to zero, as it should be in an ideal fluid flow with no waves. The dynamical vertical force as well as the moment are different from zero even at first order. The possibility f computing sinkage e.n, trim via the small FrT ex-pansion will he explored in a future work.

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bility for

-24

-(f) Stability of the Free Surface

Experiments show that as FrT increases a breaking wave appears in front of the body (Baba, 1969). The inspection of Figure 7 reveals that as FrT increases the free surface be-comes steeper. Because of the convexity of the free-surface near the body, the centrifugal effect diminishes the pressure gradient normal to the surface. When the pressure gradient be-comes less than zero, the pressure at some point inside the fluid is smaller than the atmospheric pressure. As shown by G.I.Taylor

(1950),

such a condition leads to the instability of the free-surface and very often, to its disruption or breaking. Adopting the vanishing of th:e pressure gradient as a criterion for free-surface stability, i.e., the Taylor stability criterion, we are led to the condition

u'2+v'2

pg(l

₊

_,Xt)

±

-

0

-Rewriting Equation [3.35] in dimensionless variables and with r' =

-(1+2,

)_3/2

we arrive at marginal

sta-(u2 + v2)M,xx(l+N

-3/2

-

(l+N,

2)1/2

'x' x

Expanding Equation

[3.36]

on the free-surface yields, at FrT6

order,

= - Fr 4U 2N

c'n T

-FrTG(U12N2,XX + 2U1U2N1,

- 1 + _{FrT4N1,2 + FrT6N1,N2,} = 0 [3.37]

The stability criterion as expressed by Equation [3.37] has been applied to the flow plast a box shaped body. With N1, N2, U1

and U2 given in Equations [3.28],

[3.29],

[3.33] and [3.3k] the different terms of [3.37] have been computed as functions of .

In Figure 7a we give the location of the point of minimum -P/'n as a function of FrT. The point of minimum -P/n is located at

0.3.

(g) Discussion of Results

A uniformly valid expression for the velocity and free-surface profile has been derived. The solution of first order is based on a rigid wall approximation while in the second order a singularity distribution is used to satisfy the free-surface

condition. The solution has the prQper behaviour at the stagna-tion point S (Figure 7) since both W and dN/dX vanish there in the case of a blunt body. The behavior at infinity is also

correct.

Inspection of the free-surface profile as a function of Fr(Figure 7) shows that as increases the free-surface

be-comes steeper. This is a second order effect and reflects the influence of the nonlinearity of the free-surface condition. Although at FrT of order one or larger it is doubtful whether

the first two terms represent the expansion accurately, the trend is nevertheless obvious.

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-26-The pressure gradient normal to the free-surface decreases

with

'T

(Figure

8).

At FrT

1.5,

Taylor instability of the

free surface occurs.

Of course, the value of the critical FrT

predicted by this second order theory is probably not too

ac-curate, but the calculation serves to suggest the probable

existence of a critical value of FrT, beyond which wave breaking

occurs.

In analogy with progressive free surface waves, we might

even expect the onset of Taylor Instability to coincide with

the non-existence of a free surface wave without breaking.

The present approach permits an evaluation of the influence

of the bow shape on the inception of free surface breaking as

well as the determination of the sinkage and of the trim of

bodies of finite length.

4.

High Froude Number Flow (FrT > 1): The Jet Model

(a)

General

In the case of high FrT it is appropriate to relate

the variables in the outer zone to the outer length LJ'2/gand

the velocity Ut

x = gx/G'2. y =

gyl/rJt2

.

= gyt/TJ!2; h = ghT/tJ!2.

,

The exact boundary conditions become now

(y = ii)

(y =

(y = h)

(xoo;

y.00)

with w = u - iv an analytical function of z = x + ly.

At the limit Tg/tJ"2

(Fr2

the body collapses into a line along y = 0 (Figure

3)

and the unperturbed state is that of uniform flow. The first order equations are the linearized equations of gravity waves of the type [2.18] - [2.19] (see next paragraph).

The problem of two dimensional flow, in this approximation, has been studied extensively. For the case of a blunt body at the free-surface two types of representations have been suggested in the literature:

(1) The replacement of the body by a source (Wehausen and Laitone, 1960) or by a constant pressure acting on the free-surface behind the bow (Lunde, 1952). It is easy to ascertain that the two are identical If the source is located on y = 0.

The first order velocity potential for a source of strength 0. is (Wehausen and Laitone, 1960)

+ V2 1 2 2

U -

Vfl, = 0

u-vh, =0

x u - 1; v 0

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-28-z

f.(z) =

+ ii =

z -

ezf

e {3.k3]

The free-surface profile corresponding to this solution has a wavy shape far behind the origin. It cannot, therefore,

simu-late a semi-infinite body .of arbitrary shape. Near the origin, an expansion of fi(z) for small z shows that the free-surface

is continuous there, since the integral in [3.38] behaves like Ln z for small z. The complex velocity is singular near the origin like Ln z. This behavior will be found unsatisfactory

for matching with the inner solution (paragraphs C, e).

(ii) The replacement of the body by a pressure dis-tribution singular at the leading edge like

X12

. This

ap-proach is used in studies of planing surfaces (Sedov 1965,

Maruo 1951, Squire 1957). Approximate solutions have been found for inclined flat plates of finite length by a Fourier series expansion of the pressure distribution. In these solutions the

1

velocity w is singular near the leading edge (z = o) like z

2,

while the free-surface 'is continuous there. For this reason this type of singularity, although stronger than that of (i),

is still too weak in order to permit matching with the inner ex-pansion (paragraphs c, e). An interesting feature of the planing solution is the fact that the.pressure distribution is

inte-grable. For this reason the leading edge correction and the

inner expansion are not essential. It was nevertheless assumed that a jet exists at the leading edge and Wagner (1932) has linked the jet flow and the pressure singularity in a way simi-lar to the matching of the inner and outer expansions.

Wu (1967). has studied the flow past an inclined surface in the high FrL regime by matched asymptotic expansion.

In the following paragraphs we study the flow past a blunt semi-infinite body. The outer expansions corresponds to a regime in which FrT > 1 while FrL_. 0. Hence we are in the range of displacement ships, the buoyancy being much larger than the dynamic lift, dynamic effects being important only in the bow

region.

(b) The Inner and Outer Expansions

The inner and outer expansions of Equations [3.39]

-[3i1-2] follow closely the derivations of Section 11.2, the body

length being now immaterial.

The outer expansion has the form [2.16], with =

= T'g/tJ'2 this time. Again the choice of the first order

ex-pansion is dictated by the fact that

h(x) = *H() and H(x) = 0(1)

_[3.kk]

The first order terms satisfy equations similar to [2.17]

-[2.21] which may be written in a complex form as Re (w1 + ifi) = 0 (x > 0, y = 0)

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= H

Wi = 0

-30-(x < 0, y = 0)

f1 = +

i1 and

Wi = Ui -

iv1 being analytical functions of

z in the domain y < 0.

The inner variables are those of _112.33] with Z and W

de-leted. The zero order inner expansions are exactly the

free-gravity flow Equations

[2.35] - [2.38].

(c) The Zero Order Inner Solution for the Rectangular Body

Let us consider again the simple case of a rectangular body (Figure

_9).

In the inner limit X and Y are fixed with re-spect to the bow and we assume that the free-gravity flow there takes the form of a jet directed upwards. Gravity effects are

taken into account along the free-surface upstream by the outer

expansion. The same effect on the jet upwards at some distance

from the bow is ignored.

The solution of the inner problem follows the classical methods of free streamline flow studies (Gurevich, 1965).

The complex potential F plane is mapped on the auxiliary = + ip. half-plane by

2.

d - - iT

[3. k8]

The function ç?. = £n (i/w0) = .n

(i/f w

₎ + 10 has given

where

b2 is an arbitrary constant.

The mapping of 2 on is a solution of a classical mixed problem (Signorini problem) which is reduced to a Dirichiet

problem for the function c/F. The result of the integration

we get from [3.51] and [3.52]

imaginary and real parts on the boundaries Re2 = 0 (AJ; > 0) 0 0 =7r/2 (SJ; -1 < < 0 = 3ir/2 (SB; -b2 _<

0=7r

(BA; < -ba) 0) < -i) [3.50] of Cauchy's integrals Is

fl

'1

=

-

. + ib _± iTt- _[3.51] 0 + I -

ib)J

With a new auxiliary plane w related to through

1 [3.52] 1 W 0 U)+i w-ib 2 u+ib

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Eigensoiütions of the tpe

1n+l/2

rield infinite or zero velocity far downstream.

The rnappingof the Z plane on the wplane results from the basic relationship .

z+i =

2 7T

2+ b2

dZ = dF 0 w 0 CD Z + j - w2H L CD±ib 2

wJ

u. u)+1 cn-ib -ib + (2b-l) £n (2+ b2)+o

_-ib

+ i tn

b

+(w2+ b2)2

1(J)

are ruled out since they

Substituting [3.k9], [3.52]. and

[3.53]

into Equatior [3.5k] and taking in consideration that Z = - i for w = - lb. (Figure

8)

weget

. . .

The integral of [3.55J.can_be carried out in a closed forn with the result

[3.5k]

[3.55]

[3.56]

In all the above formulae the square roots and theloga-rithni have real determination on Cl) = real.

The unknown constants and b have to be determined from matching with the outer expansion. For this purpose let us

seek the behavior of Z and W far f±om the bow, i.e. for

2Jw2

2

z + I =

L

-

1(2-b) w + (2b-l-- ) £n U) + I _{(2b-l-- )}

Similarly, from Equation [3.53] we obtain for

1(2-b) (b-2)2 i(b3- 2)

=-i-

-0 U) 2w2 3w3 + +...] [3.57]

Two cases of interest are to be discussed separately:

b ' 2. In this case in a first approximation

1 w

= -

i(2-b)2 +... [3.59] 0 7T2 = - i + (2b-1- )-2(2-b)

x+..

.. (x > 0) _[3.60] N0 = - 1 (x < a) _[3.61] 1

Hence, the velocity perturbation behaves like z 2, while 1

N

-X2.

0

b = 2. For this distinguished value

23"2

₁

wo = -1-i

3/2 z3/2

[3. 58]

HYDR0NPUTICS, Incorporated kA3/2 ₁ = '- 1 + A + 3/2 -l/2 (x > )' [3.63] N = - 1 (x < 0) [3.6'i] 0

In this case for large X the velocity perturbation decays like

z_3/2, _{while N tends to the constant value -1 + A like}

There is no ma jar difficulty in determining the zero inner solution for bodies of other shapes than the rectangular, pro-vided that e is given as a function of (for instance, a poly-gonal body). In the cas'e of an arbitra'y body wit'h given e as a function of x (or y) the 'problem becomes extremely difficlt and leads to an integr1 equation for e() (Wu, 1967).

(d) The First Order:Outer Solution

The outer problem reduces to the determination of f1(z) subject to conditions [3.k5], [3.k6] and [3i8],, while for the particular case of a rectangular body Equation [3.k7]

gives

= 1

(x< 0, y =0)

[3.65]

The problem is made unique if the singular behavior of

fi (or Wi) near' the bow (z = o) is prescribed. The inner so-lution shows that there exist two possibilities for W0:

Equa-tion [3.59] or EquaEqua-tion [3.62]. A detailed study shows that matching' is possible only in the second case. The reason is the

1

1

the origin and or (2-b) has to be of the order *2 In the first case

. is continuous at x = 0 and has the value ri = - 1

there; this requires a solution with i dropping from r1=0(x = to = - 1 (x = 0). Such a solution is not possible for the

1

assumed type of singularity of w1. If (2-b) is

Q(*2)

and = 0(1),

fli has a jump at x = 0 from (-1 + ) to (-i). Again the assumed

type of singularity of Wi does not allow for a discontinuous

fli

(see. paragraph a of this section).

Consequently, we adopt the value b = 2, and the inner term contains i as the only unknown. Moreover, w1 behaves near z = 0

3/2 1

like z / while i is singular like x 2 for x > 0.

An exact solution of f1(z)'is st±]i.difficult. The usual way to find it (Sedov, 1965) is to consider the function w1 + if1

(suggested apparently for the first time by Keldish) and to con-tinue it analytically over x > 0 in the entire z plane cut along

x<0.

With

wi(z) + ifi(z) = k(z) _[3.66]

the unknown function k(z) has to be imaginary for z = x > 0. Its

real part is in fact the linearized pressure. _{The solution of} Equation [3.66] with the radiation condition [3.k8] is

fi(z) = e

J

ek()d

[3.67]

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-36-Finally k(?) has to be determined from Equation [3.65] which yields the integral equation

x

-ix

r

1 = e _I e k(?)d7 x < 0 [3.68]

00

At this stage we do not seek a solution of Equation [3.68]

by general methods, but adopt an approximate simple expression for k(z) which satisfies only approximately [3.68].

The simplest form of k(z) imaginary along x > 0 and having the proper singularity at z = 0 is

k(z)

- 3/2

with a an arbitrary constant. From Equation [3.67] we find

z -iz a e fi(z) = i e

aJ

3/2 d? 00 -i(z+ir/k) r(-, -iz) fi(z) = ia e [3.69] [3.70]

The integral in [3.70] may be expressed by the aid of the Gamma Incomplete Function (Gradshtein and Rhyzik, 1965) and f1(z)

be-comes

and

The function r(-, -lz) Is analytical in the whole z plane cut by x = 0, y > 0. It has the following asymptotic series (GradshteIn

et al, 1965):

(i) For small z:

r(-, -iz) = r(-) + ei z

(1)n(Z)n

LL_

n! n=1 [3.72] 1

Hence with r(-) = - 27r2,

f1

has the expansion

1 _{-1w/k _i}

f1(z) = -

i2ir2 a e + 21a z 2 + [3.73] _1 .1

!1(x,0)

= -(27r)2 a + 2ax 2 + 0(x2) (x > a) I .1 = _(2)2 a + o( x 2) .(x < 0)

(ii) For large z:

-3/2 _{-13(arg z-ir/2)/2} izr .+ r(--, -Iz) ₌ e [3. 75] and = -a

e3 arg

z/2 [i + 0 [3. 76]

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-38-Again, = Imfi follows the expressions

= 0

Iz

[3. 77]

= - a + 0 (x < 0)

Izi

Unfortunately 'I is not constant along x < 0, as required

by Equation [3.65]. But the approximate solution has the proper behavior near the bow, where Wi _3/2 and i = i is like

x 2 for x > 0, and also at x - where Ill

-

a with no waves

left behind the body.

Now, it is a matter of convention how to pick the value of a in order to satisfy approximately Equation [3.65]. if we t'y to satisfy [3.56] near the bow a may be obtained from the con-dition

(o,-o) = 1 [3.78]

which gives

a = - l/(2ir) [3.79]

Although we have no exact solution for the outer problem, the approximate expression [3.71] reflects the main features of the solution.

(f) The Matching of the Inner and the Outer Solutions

The matching is generally achieved by an intermediate expansion (Cole, 1968). In the present case it can be done by the simple principle (Van Dyke, 196k): The outer limit of the inner solution equals the inner limit of the outer solution.

ia *1/2

*1/2z3/2_1+0

The outer solution matches with the outer limit of the inner solution [3.62] only if

* - 1/3

[3.81]

The estimate of

_[3.81]

is the main result of our analysis. In particular, for the value of a of [3.79],

Substituting z = in the outer solution

_[3.71]

and seeking the limit W1(Z) for 0 and Z = 0(1) we obtain from

Equa-tions [2.16] and [3.''3] for the inner limit of the outer solution

2/3

*-l/3

2

[3.80]

[3.82]

The matching of r and is also ensured at order *-1/2 with given by [3.82]: From Equations [2.16] and [3.7k] we find for the inner limit of the outer solution

1

22

N - _{1 1 + 0(1) [3.83]}

E*..

_irX2

while the outer limit of the inner solution [3.83] hasP the form

I

22 1

N

= -

1 1

+ o(*_fl

[3.8k] E

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_iO_

Bow Drag

The bow drag is evaluated from the momentum loss in the jet

D! = pU'2'

or, in a dimensionless form

D= D'/pU'2T'

From Equation

[3.81]

we have D _{FrT3 or}

D' put2T1(u2/gTf)*

[3.85]

If we assume that the bow drag for a body of finite length L' has the same expression we have for the bow drag in its con-ventional form

D'/pU'2L' = (TT/L7)(Ut/gTt) = (Tt/LI)3(2/gL?)*

_[3.86]

Discussion of Results

In the present section the free-surface flow past a blunt body with high FrT number (but low FrL) has been studied. The problem is different from that considered in planing studies, since the position of the body is fixed and its bottom is

The main results of the analysis are the following:

The proper type of pressure singularity at the. bow in the outer solution is of the order

IxI_3/2.

This pressure is not integrable so that lift may be evaluated only via the

inner expansion. Obviously, the inner solution shows that the dynamic pressure is a maximum pU'2/2 at the stagnation point.

A jet is assumed to appear at high FrT numbers. The energy of the jet is probably entirely dissipated. This is causing a drag additional to the wave resistance.

The jet thickness and the bow drag grow slowly

with FrT, like FrT1/3. .

The present analysis may be refined in different directions: By improving the outer solution, by considering bodies of finite length, by studying different bow shapes and by extending the results to real flat ships.

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-42--IV. CONCLUSIONS

Secia1 approximations are needed in order to analyze the free surface flow in the vicinity of the bow of blunt ships. In

the case of thin ships (T'/B' sufficiently small), the inner bow flow reduces to a two-dimensional flow in a vertical plane normal to the bow. Furthermore, it i' appropriate to consider a blunt two-dimensional body of semi-infinite length and this is done

herein.

The situation at the confluence of a blunt bow and the

free surface is clarified first. It is shown that there are two possible angles between a free surface and a rigid wall at the

stagnation point: (i) if the wall is inclined with respect to the horizontal at an angle larger than 1200, the free surface intersects the wall at 120°; (ii) if the wall is inclined at less than 1200, the free surface is horizontal. The latter case is usual for ships.

For small Froude number based on draft, FrT, the flow can be analyzed by means of an expansion in FrT, according to which the first approximation corresponds to replacing the free sur-face by a rigid wall. The flow past a rectangular body is analyzed to the second order. The solution to this second ap-proximation shows that the free surface becomes steeper in

front of the bow. Application of Taylors instability criterion leads to the conclusion that the stability of the free surface

In the second. approximation,, the free surface becomes unstable at FrT of about 1.5 and at a' position 30 ercent of the draft

ahead of th.e bow. The small FrT theory: does not allow the

cal-culation of bOw drag, which only, ensues after bréaking but it does permit the estimation of sinkage and trim.

For large FrT (but small FrL).thê flow past a rectangular bow has been analyzed. The problem is different

fromthatcon-sidered in planing studies, sincethe bow:is vertical, while the bottOm is horizontal. The problem is solved by matching appropriate inner and outer solutions. The ini-ier soltition cor-responds:toa free surface without gravity while the outer flow

corresponds to the'usual linearized free surface flow with gravity. The main results Of the analysis are: (i) the proper

type of pressure singularity:'at the bow ih the outer 'solution if of order

IxI2;

(ii) a spray jet appears at the bo,;whQse

energy is probably entirely dissipated This jet causes a bow drag additional, to the usual wave resistance; (iii') the jet thickness and the bow drag grows slowly with FrT, like FrT1/3.

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V. REFERENCES

Baba, E., Study of Separation of Ship Resistance Components, Mitsubishi Tech. Bul. (Japan) No. 59, p. 16, August 1959. Cole, D. J., Perturbation Methods in Applied Mathematics,

Blaisdell Publ. Cornp., 260 p., 1968.

Gradshteyn, Z. S., and Ryzhik, I. M., Tables of Integrals, Series and Products, Academic Press, 1086 p., 1965.

Gurevich, M. I., Theory of Jets in Ideal Fluids, Academic Press, 585, p. 1965.

Lunde, J. K., On the Theory of Wave Resistance and Wave Profile, Skipsmodelltankens Meddelelse No. 10, 1952.

Maruo, H., Two Dimensional Theory of the Hydroplane, Proc. 1st Japan Nat. Congr. Appl. Mech., pp. 4.09-#l5, 1951.

Ogilvie, T. F., Nonlinear High-Froude Number Free-Surface Problems, The Journ. of Eng. Math., Vol. 1, No. 3, pp. 215-235, 1967.

Sedov, L. I., Two-Dimensional Problems in Hydrodynamics and Aerodynamics, Interscience Publ., 4-27 p., 1965.

Squire, H. 6., The Motion of a Simple Wedge Along the Water Surface, Proc. Roy. Soc., Vol. 243A, pp. '48-6k, 1957.

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

Slender Ships,J. Ship Res., Vol.

8,

No. 1, pp. 15-23, 1965. Taylor, G. I., The Instability of Liquid Surfaces When

Accelerated in a Direction Perpendicular to Their Plane, Proc. Royal Society, London, A, 201, pp. 192, 1950.

Van Dyke, M. D., Perturbation Methods in Fluid Mechanics, Academic Press, 1967.

Wagner, H., Uber Gleitvorgnge an der oberflche von

Flissigkeiten, Zamm., Vol.

12, No. k, pp. 193-216, 1932.

Wehausen, J. V., and Laltone, E. V., Surface Waves,ln Encyclopedia of Physics, Springer Verlag, Vol. IX, pp.

11.k6779, 1960.

Wu, T.Y.T., A Singular Perturbation Theory for Nonlinear Free-Surface Flow Problems, mt. Shipbuilding Progress, Vol. l4,

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FIGURE 4 - FREE SURFACE FLOW IN THE VICINITY OF A STAGNATION POINT

U'

FIGURE 3 - TWO-DIMENSIONAL FLOW PAST A BODY OF SEMI-INFINITE LENGTH

(0) (b)

FIGURE 1 - NOTATION FOR STEADY FLOW PAST SHIPS

3.2-B'/V

3.0-

2.8-

2.6-

2.4-2.2 0

0

0

0

0

0

0

0Z

oD

_.

0.: ,

O ! wO

.'

&

D

Oaa

2.0 I I

0

I I 1 2 FrT = U /(gT )

FIGURE 2 - BEAM DRAFT VERSUS (FROUDE NUMBER)DJT

o CARGO VESSEL o CRUISER O DESTROYER C) PASSENGER-CARGO SHIP A PASSENGER SHIP O TUG

A

_{o CONTAINERSHIP}

o

TANKER

A

3.8

3.6-3.4.-.

HYDRONAUTICS, INCORPORATED S ----1

/

A B Iz =X+iY U=-E

O(+i)

A

FIGURE 6 - FLOW PAST A RECTANGULAR BODY AT SMALL FROUDE NUMBER

(a) (b)

FIGURE 5 - FLOW PAST A BLUNT BODY AT SMALL FROUDE NUMBER

ÀY S _A -£O A B z = x + iyI , U,

0.5

0.4-

0.3-

0.2 - 0.1 - 0.0 0.0

/

I

1 .0 e S 1' x, X =x'/T

N ='/T

N(X)= Fr N1(X)+

N7

(Jo r1 =0.5 1 .0 ..j .5 2.0 2.5

FIGURE 7 - THE FREE-SURFACE SHAPE IN FRONT OF A RECTANGULAR BODY

3.0

x

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2

Fr1

FIGURE 8 - THE RELATIONSHIP BETWEEN THE MINIMUM PRESSURE GRADIENT NORMAL TO THE FREE-SURFACE AND THE FROUDE NUMBER

I z=x

(c)

FIGURE 9 - FREE-GRAVITY JET FLOW PAST A RECTANGULAR BODY

A

A B(-b2) S(-1) 0

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Department of Engr. Mechanics University of Mechanics

Ann Arbor, Michigan 48108 Dr. Francis Ogilvie

Department of Naval Arch. and

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