Analysis of bow near field of flat ships

Analysis of Bow

_Near

Field of Flat Ships

TECIINISCHE

_UN1Vjy

°t01Um

Voor

Pthomeciii

Archlef

CD DeIft

The present ahalysis is an application of the bow-f/ow theory developed by Ogilvie (1972) to a f/at ship on the free surface. Ogilvies "bow Aéar field" is one on which a/ax = O(e -34) whereas a/a,, a/az = OV'), so that exactly the same linear free surface conditions that are familiar from the classical thin-ship theory is used. In this report flat ships are represented by a distribution of pressure on the waterplane z = 0. An approximate theory representing flow around bow is developed and compared with some experiments. The agreement is encouraging and suggests that the singular behavior of the pressure distribution at the edge may be related to the wave-breaking phenomenon around the bow.

1. IntrOductiOn

The present study is originated from the practical needs to investigate into the bow flow phenomenon of full ships

in ballast condition, which is known as wave breaking

phenomenon. The objective of the present study is to develop an analytical method for the improvement of bow fôrmin shallow draft condition.

The beam-draft ratio of full ships in ballast condition is very large, say 5. They have long parallel part after the entrance. Therefore they can be regarded as flat, slender bodies floating on the free surface. From the practical experiences it is also known that the bow form in ballast condition is quite sensitive to the resistance characteristics.

Since the body is so flat, we may replace the body by the pressure distribution on the free surface. However, in the three-dimensional case it is a tedious task to solve the integral equation which relates the body geometry to the pressure distribution. For the practical purpose it is only enough for us to analyze flow phenomenon of the most sensitive part of the body on resistance characteristics.

As the first step into this investigation, the bow flow of a flat slender, body was analyzed by applying the recently introduced "bow near field" treatment as de-scribed in the followings.

In the usual slender-body theory, we assume that

a/ax 0(1) but that a/ay and a/az=

Q(1),

where x is the longitudinal coordinate This means that rates of change in the longitudinal direction are smaller than rates of change in the transverse direction by an order of magnitude . At 9th Symposium on Naval Hydrodynamics (1972), Ogilvie presented a new approach to the slender body theory by introducing the special treatment of flow field near the bow(1). The usual assumptions of the slender-body theory mentioned above are modified in the bow region to allow for the occurrence of longitudinal rate of change greater than normally assumed, viz., he introduced the "bow near field" which is an asymptotically defined region in which

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Eiichi Baba*

a/ax

= O(e") and

a/ay = a/az =

Q(1) In this region

we

no longer have the rigid-wall free surface condition which is typical of the usual near field. Instead, we have exactly the same linear free surface conditions that are familiar

from the

classical thin ship theory.

But the

partial differential equation is the Laplace equation

in

two-dimensions, as in ordinary slender-body theory. Ogilvie presented the explicit solution of this problem for the case of a thin, wedge-shaped bow. The shape of the wave along the side of the body was computed. Some experi-mental results confirmed the prediction of the theory.

In this report we analyze the bow flow problem of a flat body on the free surface. The boundary value problem for the- bow near field of general body shape has been derived by Ogilvie. The present analysis is an application of this theory to a flat body problem.

For the convenience of understanding of the bow near field problem of flat bodies, first we cite in Section 2 the Ogilvie's derivation for the case of general body shape and in the later Sections we analyze the case of flat bodies.

2. The bow-flow problem

Let the ship be travelling in the negative x direction, the origin of coordinate being fixed to the bow. The z axis points upwards. The ship geometry is defined by the formula:

z = h(x, y),

where the non.negative function h (x, y), is the hull offset corresponding to the point (x, y) on the ship water plane. The free surface shape is given by the formula:

z =

(x, y),

defined for outside the ship.

It is assumed that the ship is "slender," which means that there is a small parameter, e, characterizing the

* Dr. Engr., Resistance and Propulsion Research Laboratory.

Nagasaki Technical Institute, Technical Headquarters

smallness of beam/length and draft/length ratios. As e - 0, the ship Shrinks down to a line, the part of the x axis between the origin and x = L, L being the ship length at the waterline. But the "slenderness" means more than this, It implies also that the Size and shape of hull cross-sections change gradually in the longitudinal direction. in particular, we shall require that:

ah/ax=o(e)

O<x<L,

even in the bow near field.

The "bow near field" is defined as the region in which:

x = 0(e"), r= (y2 + z2)

= 0(e).

It is assumed that, in the bow near field, the flow variables are changed in order of magnitude they are differentiated, according to the following symbolic rules:

ax

ar a3

az

These effect could be brought about formally through the introductiOn of new variables, x = Xe, y = Ye, z = Ze, after which we would require thai differentiation with

respect to X, Y and Z have no effect on orders of

magnitude.

Note that there is one exception to the above procedure.

We have already reqtiired that h(x, y)

0(e) and

ah(x, y)/ax = 0(e). This is simply a condition on hull geometry. It has nothing to do with the nature of a flow around the ship.

We assume everything that is necessary for the existence of a velocity potential1 which we write in the following form:

Ux + (x, y, z).

As usual, the potential satisfies the Laplace equation in the fluid domain;

[LI

0 = + +

[/e]

[chic2] [chic2]

The expression in square brackets give the order of magnitude in the bow near field of the terms immediately

above. Although we do not yet know the order Of

magnitude of cb, it is already deer that we can ignore the

term çb in finding the first apprOximatiOn to the solution

in the bow near field.

The boundary condition on the hull can be written:

o = Uh +

+ + chyhy

on z = - h(x, y)

Eel [che'"]

[0/el [0/el

2

Dropping the one term which is clearly of negligible order of magnitude, we can rewrite this condition:

H ao

02+h0

U7z

=0(e).

8n

/1+(h2

_11+(h)2

Since the operator a/an is simiiar to, say, a/ar with respect to its effect on orders of magnitudes, we can now conclude that either 0 = 0(62) or the first approximation to 0 satisfies a homogeneous boundary condition on the hull. Let us suppose that the former is true.

if this is

wrong1 we shall discover that fact when we consider the other conditions on 0.

There are the usual two boundary conditions to be satisfied on the free surface.

[Al

[s']

[%]

[e3] [2] [e2)

onz ='(x,y).

[B]

[/e"]

[efl

[i']

[ci

The orders of magnitude involving 0 have been noted1 but of course we have not yet reached any conclusions, even tentatively, about the order of magnitude of '. In

condi-tion [A] we can clearly neglect all

of the quadratic terms, and in condition [B] the second and third terms on the right side can be neglected. Thus we have reduced the number of terms to the following:

0 = g' + Ucb,

0 = U

-In [A I, the first term cannot be lower order than the second, because we would then have the meaningless

results: 0. Thus, either the two terms are the same

order of magnitude or the first term is higher order than the second.. If the latter is the case, the first term in [B I is higher order than the second term in [ B I, and this leads to an ill-posed potential problem. Therefore we must conclude that

' = 0(e), and the two conditions are

consistent in order of magnitude. Note that this order of magnitude estimate for allow us to impose the boundary. conditions at z = 0 with negligible error.

Finally, we can combine the two conditions above into the following:

[F]

+ , 0z,

on z = 0, where ic = g/U2.

In finding the first approximation to 0, we have a

boundary value problem to solve in the y plane. That

I

is, we have a partial differential equation involving only the transverse rates of change. The body boundary condi tion is a simple Neumann condition but the free surface condition involves derivatives with respect to x; and so a 3-D aspect is introduced through this condition. The

problem in the cross plane is illustrated in Fig. 1 ( a ).

(a) General body z = -h(x,y)

(b) Flat body : = -h(x,y)

Fig. 1 Problem for the first approximation

3. FIat body prOblem

In this section we shall confine our attention to a special case of this problem, namely, to flat bodies which can be represented by a distribution of pressure on the waterplane z = 0. This special case 'is depicted in Fig. 1 (b).

For the flat bodies being considered we shall set the following boundary value problem for the bow near field:

EL]

z<0

'[F]

+,= 0,

z' = 0 for Iy.I >S(x)

[B]

Uh,

z =0 for IyI<3().

In order to solve this boundary value problem, we begin with a general expression for the solution which satisfies the free surface boundary condition Outside the region S which is the projection of the wetted surface to the

plane z = 0 Let us define a function ti(x y z) by the

equation:

(1) This is a harmonic function in the fluid domain so that

+ Iizz = 0.

(2)

= ± x 0

iJ, vanishes at infinity and also on the plane z 0 outside S because of the condition [F] 4i is supposed to be finite

on S. Let us designate the value of on S by

i1(x,y).

These conditions can be satisfied by a double sheet

pOten-tial such as

i(x,y,z) =

_(yy')2+z2

dy(3)

Now let the Fourier transform be defined as f011Ows:

Ø*(y z)

_{=f:x,yfze_ith,}

(x,y,z)

=

f*(wyz)UXdw

'Taking into account that and its derivative vanish

at 'infinity, we have the following Fourier transformation of the equation (1)

2i(c.,,y,z)

+Køz*(I),y,Z).

₍₄₎

A particular solution of the differential equatiOn (4) is w2

r

w2

ø*(wyz).=!ek

_{j e"}

which must be added by the complementary functiOn x*(x, y, z) that is a solution of the homogeneous equatiOn

*+gX:.=O

_ônz=0.

Then the inverse transform of Eq. (5) added by*(,y,)

(x,y,z) =

ifxdf(cYu)du

"-i- x(x,y,z),

' (6)

where x (x, y, z) is the inverse Fourier transformation of x*(w,y, z) and is a harmonic function in the fluid domain,

Xyy + Xzz '= 0 satisfying the boundary conditiOn

+

_{Xz = 0, on z}

0.

In order to express the velocity potential in term of

ii (x,y) we introduce

the integral representation in,,

Eq. (3) as follows:

(y_y )2+ z2

I ëcos j3 (yy') d.

(7) Using the Fourier integral theorem, we obtain from (6)

(5)

z

0

i:'i:

(x'.')dy'f

cos1f3(y-y')Jexp[3z+k.,(x'x)]

_+x(x,y,z).

(8) 2 Denoting a7(x' y')

iP

(x',y') =

_{-, a} (9) On the other hand, three-dimensional solution of our

X _{problem can be simply considered as follows. When we}

and Integrating by parts we have look at a slender flat ship far from the body it can be represented by a line pressure distribution on the free

Ø(x,y, z) =-

surface. The three-dimensional solution for the case of

2,r general distribution of pressure p(x,y) on the free surface has been derived for instance by Wehausen and Laitone cos (3 (yy') I exp [$z+k,,(x'x),J (1960), as f011oWs2:

X

- 2

-C,)

1

i'

i'

Ø(x,y,z)=

₂

J dx'J dy'p(x',y')/ dOsecO

Since there is a relation:

ir PU -

'0

,

,exp[iw(x'x)1

P.v.I.

.dw

J''

(i) '(13

= sgn(x'x)iricos/(X'Xfl,

y, z) is rewritten as f011ows: 1 ,'x pS(X')

cb(x.Y1z)=--j dxj

'y(x',y')dy'J

d(39Pz

2ir-

-s(x') .0

where it is assumed that 'y(x, y) vanishes outside S This is the solution of the bow near field of flat ships. However, the functions X(x,y,z) and 'y(x,y) are not determined yet.

4 Matching with the oUter sOlution

Behavior as y -*oo is an important limit, for it provides the cOnnection to the three-dimensional solution. Our solution in the bow near field, when we lety eo shOuld

match the solution of the three-dimensional problem if we let x, y, z 0 in the latter. x(x, y, z) is determined so as -to match the both solutiOns.

FrOm (11) we have, the f011owing expansion asy-oo':

x cos (j3y) +

:'_f'a:'

I'(x')

f';i13

e5Z

cos kj(x'x )}

2ir 0 where

i'

kelczdk x P.V. I J0

seco/c

xcOs((3y) +x(x,y,z),

(12)

x (x,y,z)

= lim x(x,y,z),

.s(x')

rv) =f

'y(x',y')dy'.

-s(x')

L:uL:dy'Pxu,y')Ldo sec30

e"2

x cos['((x'x) secO] cos[(y'y)

sec2OsinO]. (13)

We are therefore able to derive the potential function for the case of line pressure distribution by letting y' = 0 as:

x cosIv'W(x'x)I cosU3(yy')I

(x,y,z) =

x'fio secO

1 ' s(x')

e(3Z

+ ff

y )df d(3

x p n

kedk

sin[k(xx )cose]cos[kysinO] J0

gsecO-ccosIV(x'_x)IcosI1Y_Y')I+X(X,Y.Z),(1l)

4x,y,z) ---

_f XdxI(x)f

4(3 ecôsI /j(x'x)I

x

----

f(x')dx' f'e sec3O e

2Ocos['((x_x) secO] irpUJ_., J0

'x cOskysec20 sinG], (14) where

P(x') =fP(x')ldY'.

In order to obtain the inner expansion of this potential function. first we rewrite the integral with respect to 0 and k with variabl of rectangular coordinate system:

2

rP(x)dxv.f4fd

w-pUJ... 0 0

(x,y,z)

-x sin [k(-x--x') cos0]cos[k(y'y) sinG]

exp[/w2+132 z]wsin)o.,(x'xflcos3y

- ,/ w2

+132

1

fP(x'kii-. j?zcovi-xn

Then we introduce the following new variables:

X=xe", Y=ye1,Z=zC'

n =1k ,P(X) =P(x)e.

Using these variables, we have the expression for in

terms of X, Y, and Z:

Ø(x,y,z) =

(X, Y,Z) = 2

rP(x')dx' p.v. r

rn

irpUj

Jo _'4 \/ em+fl2Z e _{msunlm(X'-X))cos)nYI} m2-i.,/ em2+n2 2irpU

fP(

X')dX'f ?Zcos

(X'-X)}

x cosk/v'n-e YI

As e -+ 0, the first approximation of the inner expansion of 0 is obtained as

(x,y,z) =

(X, Y,Z)

P.V.fdmfdn

eZmsinm(X_X)} cos)nY

x m2 -Kfl

7puf'(

dx'f e?Zcos

x cosnY)dn.

Ø(x,y, z) ff2pU

JP(x)dx

eosin1ci.,(x'-xfl

x P.V.fd(4)fdi3

,2iq3

cos(3y) =Wsin

_ki.(x'-xH

d 0

= sgn(x'-x)--cos/(x'-x)}

finally we have the

first approximation of the inner

expansion of the outer solution as follows:

0 (x, y, z) -

f')

f'sz

cosI3y

xcosfZ(x-x'fl df3

1jP(x')

frf

az

_{'/x-x') dj3}

2npU

__i_ r°x')

2irJ_.,pU

1

fXP(X)

dxf ei3Zcos13y_{cos I}

_%Jx-x')

_{I d13.} pU

Matching this expansion with the first approximation of the outer expansion of the bow near field solution (12), we have the following relations:

r(x)

=P(x), i.e;fs(X;((x,y)dy

js(x)P(xY)

dy,

pU -stx) .-s(x) pU

xjx,y, z)

= -

dx'f?cosI3Y

x cosIv'(x-x'fld$.

(20) Thus it is easily supposed that the function x (x, y, z) of the bow near field may take originally the following form:

(16) 1 s(x')

x(x,y,.z) =

_{--fdx'j 'y(x',y')dy'}

fe

2ir"_,. _-s(x')

Substituting (21) into

(11), we finally

obtained the following solution for the bow near field problem

1 x s(x'}

O(x,y,z) =

_fdx'f y(x',y')dy'fe

'-s(x')

1

f(x')dx'f?cosI./(x'-x)}cos13Ydf3.

(17)

xcosl3(y-y')I cos lZ(x-x').dj3,

(22)

2irpU-

0

5

xcosIv1y}

d3. (15) Using a relation:

Rewriting this expansion in terms of the usual variables,

we have _x

cos{13(y-y'fl cosIsJ(x-.-x')} d13.

(21)

dn V1

fl-where 7(x, y) is determined from the body boundary condition

ø(x,y,z) = Uh(x,Y),

on z = 0.

This boundary condition gives an integral equation for (x, y):

1 x

Uh(x,y) = _fdx'f7(xy')dy'

The velocity potential expressed by (22) is identical with the velocity potential for the low aspect ratio planing surface derived by Maruo. It is the main difference in the derivation that Maruo determined the complementary

function x from the radiation condition while in the

present analysis it is determined by matching the solutions of bow near field and far field.

It is characteristic of the first approximation in this

theory that a disturbance at a particular section can never have an effect upstream of that section. This suggests that the flow around the bow is largely independent of total ship length, and sO it should be possible to predict the bow flow independently, of the flow around the rest of the ship. Therefore for the analysis of bow region we can expect the existence of a local characteristic length, i.e., entrance length or beam, besides the ship total length. the Froude number based on such characteristic lengths is much higher than the Froude number defined by using the ship total length. Thus we can easily accept the fact 'that the solution of the present analysis for the bOw near

field of flat ships coincides with that of the high-Froude-number low-aspect-ratio planing surfaces even when our flat ships have long parallel body after the bow.

5. Approximate solution of the integral equation

In order to solve the integral equation (23), we employ a functiOn. defined by

;i(x,y)

=

_-f7(x',Ydx'.

(24)

Integrating by parts the equation (23) and using the relation

;-(

=Jicos

113(y--y')I l3,

we have

1r-)

a 1

h(x,y)

=

-J

ir _-sCx)

z(x,y')(

ay 'yy

)dy'

x s(x')

-f d4 p(x',y')dy'f

_ir

-

-s(x) 0

x jl3cosII3(yy')I cosk,/'(xx')} d13.

(23) X IogI

s(Y)

1d

we multiply

'ii ( Y, y)

x cos_113(yy')} sin

Ji(xx'fl d13.

(25) Since there are the following relations:

1 a 1

w()

₌

-f u(') (

,')d',

ir a

u(Y)

=--fw(n)

1

s(x)2Yy+

..,/s(x)2_Y2s(x)2_y

=logl

_s(x)(yY)

on the both sides of equatiOn (25) and integrate with respect to y. Then we have

s(x)

(x,y)

=

f h(x,y) 'I'(Y,y)dy

_-s(x)

/

,,x ,.s(x')

+ V

J

_-

dx'J

_-s(x')

(x'y') K(x, Y,x',y')dy',

(27) where 1 ' s(x)

K(x, Y,x',y')

=

-f F(xx',yy') 'I'(Y,y)dy,

_-s(x)

F(xx',yy')

=

fi3

cos/3(yy')

sinf(xx')dfl.

When we look at the leading edge regiOn between x = 0 and x = I where I = 0(e')'and 'a >0, even in the moderate speed range, viz., 1/ic = U2/g = 0(e), the second term of (27) becomes higher order than the first term by O(e°). Then the first approximation to .z (x, Y) is given as follows:

pS(X)

,L(x,Y)J

h(x,y)"P(Y,y)dy,

(28) -s(x)

which is free from the gravity effect. In other words, it may be said that the gravity effect can be neglected at the leading edge region of the flat ship even in the moderate

speed range, say U/./i7- 0.3 . In Appendix A results

of some numerical

studies on the gravity effect are

described. From this study it is indicated that the gravity effect on the drag and lift is small in the leading edge

region when x is

less than 30% of Ship breadth and

Froude number defined by U/'.,/ exceeds 0.5 (for ship ofL/B = 7,

U/..,fE = 0.188).

6. Effect of leading edge form on drag

As it is shown in the previous section that the gravity effect is small in the leading edge region, we use the simple relation (28) for the study of the effect of leading edge form on drag instead of solving the integral equation (27) completely.

The drag of the body between the leading edge and a certain point can be expressed by

x s(x')

D(x)

=

f±'f p(x',y')h.(x',y')dy',

(29)

0 _-s(x')

and the lift of the part of the body is expressed by ,,x s(x')

L(x)

₌

_J

dx'f

p(x',y')

dy'.

(30)

o -s(x')

As an example, a flat ship with a parabolic bottom shape is considered here. Fig. 2 shows the cross section of the body.

In order to obtain j.i(x, Y) from Eq. (28), we introduce new variables as follows:

y = s(x)

cosO, Y

= --s(x)

cosO'.

Since '4'(Y,y) in (28) is expressed as

4'(Y,y)

= 'I'(O',O)

='logl

sin (0-0')

2

2=

1

= -

_{T n1 fl}

sunn0sinn0,

we obtain the following relation: 2s(x) 1

z(x,O') =

-

sinnO'

ir

,=1fl

X

fhx(x, 0)

sinnO sin0dO. (31)

s'(x)

h(x,0) =(1+cos20).

£2 S(x) Q: constant S(x)

B-S(x)/.Q Zh(r,y) Iti'S(x)'I BS(x) y

Fig. 2 Flat body with parabolic bottom

Substituting this into (31), we have

2s(x)s'(x)

5 1

.z(x,0

= 8

sin0 sun30

I.

Then the pressure is expressed as

p(x,0')

= pUy(x,0') = pU2ii(x,0')

2p'J2

[

(s'(x)2 + s(s)s"(x)l

--

sinO'+-- sin30'}

5' cos20' 1 cos30'cos0'

+s'(x)2

+

-8 sinO' 8 sinO'

Finally we have the following expressions for the drag and

lift:

PU2 119

2 9 D(x) = I

{s(x)s'(xH

+

_rS'(X')3S(X.).].

cz2 8J0

pU2 5ir

L(x) =

s(x)2s'(x).

£2 8

It should be noted that the pressure becomes infinity at 0 = 0 or ir, viz., at the edge of the ship. Singular behavior in the linearized theory suggests that there are non-linear phenomena, like breaking of waves, around bow. A con-sideration of the singular property of flow around bow is given in Appendix B.

If the water line form s(x) is given, we can calculate the pressure distribution on the ship surface, the drag and the lift by (32), (33) and (34) respectively.

In order to study the effect of leading edge form on the drag, the following three water line forms are compared:

Wi:

s(x)"axa2x2/2B, 0xB/,

W3:

s(x)=-13x+--B,

3 2 2B 2B

=flxj32x2/2B,

0 0

where a= 2tan2O , 3= 2tan3O and B is the beam of

the body. Fig. 3 shows the comparison of the water line forms. Although we are considering the leading edge region of the body, for the convenience, we calculate the drag of 'the each body between x = 0 and x =' x0 where s (x) = B/2. Thus the water line reaches the beam of the body. The calculated values of drag are as follows:

W2: s(x)=Oxa2x2/28,

B/3,

Wi W2 W3

1 0.01734 0.04363 0.03219

for=3

Comparing Wi and W2, we see that, finer body Wi gives smaller drag than the full body W2. The body W3, which is designed by attaching the bow projection is evident in reducing drag. Those theoretical results coincide with our practical experiences.

8

D (x0)

OWL

Fig. 3 Comparison of water line forms

7. Experiment

In order to determine whether the theoretical results are even approximately valid, some experiments were

conducted with the three flat

hips described

in the

previous section, which have long parallel parts after the entrance. The beam/length ratio of the models is 1/7.

For the study of effect of bow form on the drag, each model has the same parallel part and after body. Fig. 4 shows the body plans of the models and Table 1 shows the particulars of the mOdels. M.2116, which corresponds to the water line form Wi is symmetrical to the midship. Fig. 5 is the pictures of this model. M.2117 corresponding tO W2 and M.21 18 corresponding to W3 have the same after body as M.2116.

The items of experiments are as follows: Towing test

Wake survey at 0.5 L behind A.P Wave pattern measurement

Pressure measurement on the body surface

Flow measurement near the leading edge by use of 5-hole pitot tube

The towing tests were conducted in the speed range

AP. 1/4 1/2 3/4

Designed water line

F.P. o

M. 2118

M2116

Fig. 4 Lines of M.2116, M.2117 and M.2118

0.15 < U/f

< 1.0 (0.06 < U//E <0.38)

Fig. 6 shows the comparison of the residual resistance coefficients by use of Prandtl-Schlichting formula of fric-tional resistance. The difference of bow forms clearly reflect on the difference of drag as shown in this figure. The fullest ship M.2117 gives highest values and the finest ship M.21 16 gives smallest values in the speed range

U//

> 0.5. The effect of the bow projection in reduc-ing drag is also shown by M.2118.

For the comparison of experimental results with theo-retical ones, the effect of the parallel part and after body on the drag should be eliminated. Since the present models have rather long parallel part, we assumed that the influence

of the after body on the drag of the entrance part is

negligibly small, i.e. we assumed that the difference of residual resistance of two models can be compared approxi-mately with the difference of pressure drag of the cor-responding two models derived by the present theory. Fig. 7 shows such comparisons. The difference of the residual resistance of M.2118 and M.2116 in the Speed

range LJ/../i > 0.7

is about half of the difference of theoretical values. On the other hand the difference of the residual resistance of M.2117 and M.2116 gives a little lower values than theoretical one in the speed range

Table 1 Particulars of models used

M.2117 2118

l

F:P. M.2116 M.2117 M.2118 Lpp (m) 7.000 7.000 7.000 7.000 7.000 7.115 B 1.000 1.000 1.000 d 0.1667 0.1667 0.1667 V (m3) 0.6353 0.6617 0.6627 Cb 0.5445 0.5671 05680 Cp 0.8168 0.8506 0.8520 Cm 0.6667 0.6667 0.6667 L,,,/B 7 7 7 Bid 6 6 6 Le/Lpp (entrance) 0.1963 0.1237 0.1237 (run) 0.1963 0.1 963 0.1963 8.805 8.809 8.838

U/-../k > 0.7.

It may be said that the agreement of order of magnitude of drag values in the range of higher U//T values supports the present theory, since our theory is

Fig. 5 Model M.2116

0.06

0.04

0.02

0.0

developed assuming

U2/g = O(),

i.e.

U2/gB = 0(1).

In order to study more on the flow phenomena around the bow region of the flat bodies, we carried out the flow observation, wake survey and wave pattern measure-ment.

Fig. 8 shows the comparison of flow phenomena around bow. In this figure spray like phenomena 'are observed. The drag due to this phenomena can be detected by means of wake survey. Fig. 9 shows some results of wake survey conducted at the control surface 0.5 L behind A.P. of the bodies. The central peak of total head loss distribution is due to friction on the body surface. The side peaks cor-respond to the momentum loss due to spray like phenome-na at the bow. As mentioned before the pressure distribu-tion derived by the present theory diverges at the edge of the body. Therefore it may be supposed that high pressure distribution at the edge of the body causes such spray like phenomena. It may be also said that the drag derived by the present theory gives the drag due to such spray like phenomena, which is rather similar to the wave breaking phenomena around blunt bow of full ships in ballast condition.

Figs. 10, 11 and 12 show the comparison of the resistance components of each model. It is noticed that the drag due to spray like phenomena, denoted by CL, is much greater than the wave pattern resistance C,,. Fig. 13 shows the comparison of the theoretical values and the experimental values of drag due to spray like phenomena. The experi-mental values tend to increase with Froude number. Ap-proximate agreement of orders of magnitude is observed.

In order to confirm further the validity of the present theory, the pressure measurements were carried out. Fig. 14 shows the arrangement of static pressure holes on the surface of the body. Fig. 15 shows the one of the results compared with theoretical values at = 0.80. A

I

/

/

0 M.2117 ,' M.2118, 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 6 Comparison of residual resistance

9

0.06 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.38

10

M.2115

M. 2117

M.2118

Fig. 8 Comparison of bow wave breaking

0.2 M. 2118 0 F.P. M. 2117 M.2116 _______k

-:

qualitative agreement is observed

experiment the flow measurement a,5-hole pitot tube was conducted. B flow property predicted by the ported by the experiments.

Designed waterlines (x)

In addition to this

around bow by use of As shown in Appendix present theory is

sup-E..=0.65 h.= 0.65 M.2116 F. =0.65 Theory (M.2117-M2116) 5xpeiment.. Cr M.2118-Cr M.2116

Fig. 7 Comparison of difference of residual resistance

(110-H)

,

Depth 100 rns,.from tree surface M. 2117 F=0.55 -1.0 M. 21 18 50 F,.0 =0.65 I . L,.iIMPl)v... 1.0 1.) 0.01 -1.0 _-0.5 u 0.5 Model breadth-I

Fig. 9 Comparison of head loss diStribution

0.04 0.03 0.0 M.2116 CsS ---ITTC 1957 Hughes

Fig. 10 Resistance components of M.2116

(H0-H) _{lead loss due to}

wave breaking at bow Cr M.2117-Cr M.2116

/

Theoi (M.2118 -M.2116) ExperiTnent 0.6 0.7 0.2 0.3 0.4 0.5 F0 F.= 0.65 (H0-H) 1000

0.04 0.03 0.02 0.01 M.2118 M.2117 0.0 0.2 0.3 0.4 0.5 0.6 0.7 FuR

Fig. 12 Resistance components of M.2118

Detail of Static pressure hole Model surface p18 c'v c_ + C,..k. CR, 1957 - - Hughes + Cm', C,.R,.k. ITTC 1957 Hughes

Fig. 14 Static pressure holes To manometer Static pressure holes Turbulence stimulato Static pressure holes 0.5 0.6 0.7 0.8 F

Fig. 13 Comparison of predicted values of drag with experimental values

Fig. 15 Comparison of pressure distribution theoretical and experimental on the model surface of M.2116at F,, = 0.80 11 18 M.2117 0.04 Theory 0 Experiment 0.03 M.2116 0.02 0.01 0 I I I I 0 0.5 0.6 0.7 0.8 Theory 0.04 0.03 M.2117 0 U 0.02 0 Experiment 0.01 0 0 0.5 0.6 0.7 0.8 Fu 0.04 _M.2118 Theory 0.03 0.02 0.01 0 Experiment 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 11 Resistance components of M.21 17

0.07 0.06 0.05 C) 0.04 0.03 0.02 0.01 0.0

8.. A leading edge form of minimum drag

There is a practical interest to find a leading edge form Of minimum drag. Fortunately the drag is expressed in terms of water line form s(x), for instance, as

pU2

r s'(x)3s(x)cLx (35) D(x0)

= cz2

8 "o

for the bodies cohsideréd in the previOus sections:

So it is possible for us to apply the method of calculus of variations in finding the minimum drag form.

Let the following two boundary conditions and one constraint for s(x) be introduced for the calculus of variã-tions.

s(0) = 0, s'(x0) = 0,

I.

(3)Js(x)dx = constant (given value).

0

Then we have a differential equation fOr s(x) from Euler's equation as:

s'(x)3 + 3s(x)s'(x)s"(x) =

A (36)

where A is the Lagrange's multiplier. The solution of the differential equation is given as:

s(x0)a

iT

,x

J--

s(x) s(x)

+s(x0)I

where

A in (37) should be determined from the constraint (3). In practice, however, since we obtain the relation

3/A

2ir

x0 /

--

3V' s(x0)

when x ± X0 in (37), A can be determined, if the ratio x0/s(xo) is given.

Fig. 16 shows the comparison of the. minimum drag form derived by (37) and the form of M.2117. It should be noticed that the minimum drag form has a blunt water line form.

Fig. 17 shows the experimental result for this minimum drag form comparing with M.2117. As the theory predicts, the reduction of drag is attained by the minimum drag form. The magnitude of drag reduction by the form is 12 I = log arctan

s(x)[1_(1_)'/3]

I s(x0) 1+211 s (x) S(X0): (37)

about 0.005 in the speed range U/.,/ > 0.65, while the predicted value of the magnitude of reduction is 0.00515. This experiment also confirms the applicability of the

present theory to the analysis of bow near field of flat

ships. X0/B= 0.8660 0.5 M. 2117 0.4-0.20 0.30 0.40 0.50 Mm. drag form m 0.3 C1,1=0.04363 M.2117(B/d=6)

/

/

=0.03848 Mm. drag form 0.2 _(B/d=6)

/

/

/

/

0.60

Fig. 16 Water line form of minimum drag

0.8 1.0

0.70 0.80

Fig. 17 Experiment of minimum drag form

0.90

9. Concluding remarks

As described in the previous sections, the experimental studies suggested the validity of the present way of ap-proach to the bow flow problems. For practice the present theory can be served as one of the tools to analyze the resistance characteristics, of the bow form in sh3llow drft condition.

At this moment, there is no satisfactory explanation of the reason for the occurrence of wave breaking phenome-non around bow. The present theory suggests that the singular behavior of pressure distribution at the leading edge may relate the phenomenon. Further studies are anxiously awaited.

Acknowledgement

The author wishes to exiiess his deep appreciation to Professor T. Francis Ogilvie of The University of Michigan, who motivated him to look at the present problem as an example of problems which can be tackled by the method of matched asymptotic expansions. The author also wishes to express his thanks to all members of Nagasaki

Experi-mental Tank for their cooperatiOn in carrying out this investigation.

/

M.211/

/

Mm. drog

/

/

/

/

0.2 0.4 0.6 0.1 0.04 0.03 II 0.02 LI 0.01

(1) Ogilvie, T. F., The Wave Generated By a Fine Ship Bow, 9th Symposium on Naval Hydrodynamics, Paris (1972).

(2). Wehusen, J. V. and Laitone, V., Surface Waves, Handbuch der Physik, vol.9, Springer-Verlag, (1960)

A. Gravity effect on the lift and drag

The integral equation (27) in the text is the Volterra type integral equation. Denoting the first approximation (28) by Mo(x, Y), the next approximation to j.i(x, Y) is

given as:

-p1(x,Y) =u0(x,Y)

ir

x s(x')

f_dx'f ji0(x',y')K(x,

Y, x',y')dy',

(A-i) _

- -s(x')

tion of gravity effect on the lift and drag acting on the leading edge part of the flat, ship.

For the numerical study a flat ship expressed by the following formula is considered.

i0(x, 0') 2s(x)s'(x)

: sin0' + --sin30'

, (A-3)

where the variable 0' is introduced through the relation

Y = s(x)cos0''

Differentiating (A-i) with x, and substituting J.L1(x, Y) into the following expressions for lift and drag

,x

,,s(x')

L(x)

_{= J}

dx'J

p(x', Y)dY

0 .-s(x')

x s(x')

=pu2fdxf

iL.(x', Y)dY,

0 -s(xl

x'

s(x')

D(x)

pU2fdx'f

z.(x', Y)h.(x', Y)dY,

0 -s(x')

References

Appendices p.598.

(3) Maruo, H., High- and Low-Aspect RatiO Approxima tion of Planing Surfaces, Schifftechnik (1967) Bd.14-Heft 72, p.57.

the expressions including gravity effect are obtained as:

L(x)

ir 5

2'

=

_j_j[s(x) s(x)]

+'---j"t's(x') fdr s(t)s'( r)

cos /(x' t)}

fd3Ji(13s(x')H--Ji(13s(t))

where = g/U2

D(x)

iT 119 2 9

This expression was used to get the quantitative' informa- pU2 = 1Z2[48

x)

+ j-fs'(x')3s(x')dx'

+

-2f dx's(x')s'(x')f dts(t)s'( r)

cos (\/j(x' t)1

(A-4)

:i

(A-4) and (A5) coincide with the expressions (33) and (34) in the text.

Figs. 18 and 19 show the calculated results of lift and drag versus FflB

U//

for the water line form s(x) expressed by the fOrmula:

s(x) = axa2x2/2B,

a = 2tan2O°, B: Ship breadth, (A-6) d:'Ship draft

The' calculated results Show that the gravity effect is small in the leading edge region where x is less than '30% of ship breadth and Froude number U// exceeds 0.5.

B. Flow characteristics around bow

From th.e numerical, study (Appendix A) it was found 13 '0 1 J2(3s(t))

[_Ji(s(t)

z=h(x,y)

=

y2s(x)l. 2=cont.

(A-2) 2 13s(t) x _(13s(x')) 1 J2(13s(x')) (A-5) 4 2 as(x)

]

This form is used also in the section 6 of the text and the first approximation ji ( x, Y) is obtained as

that the gravity effect is quite small in the leading edge region. Based on this fact the velocity components at. 'flB '°° are calculated for the ship form expressed by (A2). From (22) in the text we have

ia

uIUr---=

_Uax.

u/U= -!7-:.-

:th'fos1h y)G(

dj3,. (B-2)

ia

w/U=--

_Uaz

where G(a. x) = )s'(x)2+-s(x)s"(x)) 2 3s(x) +s'(x)2 I -13s(x)Jo(t3s(x))

_!_1

(13s(x)) .3J(I3s(x)) 2 j3s(x)

For the numerical study the water line form s(x) expressed by (A 6) was used Fig 20 shows the comparison of

velocity components at (x = 02 B y = s(x)) with the

measured results by use of a spherical .5-hole pitot tube of

7,mm for ship model M.2116.

It

is characteristic' of flow arpund. the bow that the.

velocity components show the depthwise rapid change near the free surface The predicted property of flow near the free surface agrees well with the experiments

To understand flow characteristics near the free surface the velocity components at the free surface (z 0) are calculated from (B-i), (B-2) and (B-3) as follows:

14

0.15

Q10

- 0.05

F = uII

Fig. 18 Gravity effect on lift

F

F05

= 1.0 F,, 0.1 0.2 0.3 9.4 0.5 0.6 0.7 0.8 E=X/B

if-Lcos(y)Gu3,x)d., (B-i)

fecos(f3y.)Gu3..x')da. (B-3)

0/UIz=o u/UIz__0. = 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 = X/B

Fig. 19 Gravity effect on drag

-0 .

iy

I> s(x)

2

--s'(x) s(

s'(x)2 s(x) ---.s"(x)I q/s(x)2=y2

11s(x)

2s'(x)2 - -'2 s(x) s(x)2

ls(x)2y2I ,

IyI <s(x)

0, lyl > s(x),

1 JS(X)2 y2 3

')- y s'(x)

_/2.

_{- 2)}

22sJy2-s(x)2

&1 s(x)'"'

s(x)

-s'(x), IjI

wIUIz=0 =

2iL

,

IyI<s(x);

ç2

body boundary condition

It

is evident that the velocity cornponeht haè the

singularity at the edge of the body [y

s(x)] This

singular property in the linearized theory indicates that there exist nonlinear phenomena around bow.

,

V(X)2_y2

;

s.Js(x)_y2

U-0 u/LI .L

/

Uau L Theory-. fTheory

I

Measured at u=0.2 yS(a)+35mm(M.2116)

-0.3 -0.4 -0.5 -0.6 tJ/v=0.3 Measured 0.5 o 0.7

Fig. 20 Comparison of measured velocity components with theoretical values at poiflt A (x = 0.28, y = S (x))

0 a/B 15 (1+e/IJ) 1.0 -0.1

(1+)

Theory at ,j=S(o) - -0.2