An application of linearized theory to water entry and water exit problem. Part 2 (With ventilation)

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NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

Washington, D C. 20034

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AN APPLICATION OF LINEARIZED THEORY TO WATER /F*i.--.

ENTRY AND WATER EXIT PROBLEM PART 2 (WITH VENTILATION)

by B. Yim

This docunient has been approved for public release and sale; its distri-bution is unlimited.

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July 1970 Report 3171

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Washington, D.C. 20034

AN APPLICATION OF LINEARIZED THEORY TO WATER

ENTRY AND WATER EXIT PROBLEM PART 2 (WITH VENTILATION)

by

B. Yim

This document has been approved for public release and sale; its distri-bution is unlimited.

TABLE OF CONTENTS Page ABSTRACT 1 ADMINISTRATIVE INFORMATION i INTRODUCTION 1 FORMULATION OF PROBLEM 3 CONFORMAL MAPPING S

GENERAL TECHNIQUE FOR OBTAINING SOLUTIONS 6

WATER ENTRY OF A NONSYMMETRICAL WEDGE 8

WATER ENTRY OF A FLAT PLATE WITH LEADING EDGE CAVITY 10

BASE-VENTED SYMMETRICAL WEDGE 13

DISCUSSION 17 ACKNOWLEDGMENTS 19 REFERENCES 28 LIST OF FIGURES Figure Figure Figure 1 2 3 - Nonsymmetrical Wedge - Ventilating Flat Plate

- Base-Vented Symntetrical Wedge

P ag e

20 21 22

Figure 4 - Pressure Distribution on the Pressure Side of a

Nonsymmetrical Wedge Entering Water 23

Figure 5 - Pressure Distribution on the Suction Side of a

Nonsymmetrical Wedge Entering Water 24

Figure 6 - Effect of Changing Angle on Pressure Distribution on

the Pressure Side of a Nonsymmetrical Wedge 25

Figure 7 - Pressure Distribution on a Vented Flat Plate Entering a

Water Surface 25

Figure 8 - Drag and Lift Coefficients of Nonsyrnrnetrical Wedges and

Ventilated Flat Plates Entering Water 26

Figure 9 - Pressure Distribution on a Base-Vented Thin Wedge

Entering the Water at F =

r 26

Figure 10 - Drag Coefficient of a Base-Vented Symmetrical Wedge

Page

Figure 11 - Cavity Shape of a Ventilated Flat Plate Entering

Water

27

Table i - Singularities of w and w

NOTATION A0, A., A I Coefficients to be determined

A,B

f C Chord length D Drag

K Given in Equations (10) and (13a)

p Pressure

t Time

u x-component of velocity of water particle

V Foil speed

y y-component of velocity of water particle

w Complex velocity

x, y Rectangular coordinate system

z

=x+iy

0 a2 Wedge angle

a Angle of attack of flat plate

Transformed coordinate

p Density of water

Potential

Subscripts t, x, y, and designate partial differentations with respect to t, x, y, and , respectively.

*

References are listed on page 28. ABSTRACT

A conformal mapping technique is used to analyze

two-dimensional thin foils entering a water surface vertically

with ventilations. Boundary conditions are linearized, and the gravity effect is neglected. Flow fields, pressure

dis-tributions, and drags are given in closed forms for an in-finite nonsymmetrical wedge, an inin-finite flat plate with

ventilation from the leading edge, and a finite base-vented

symmetrical wedge.

ADMINISTRATIVE INFORMATION

The work reported herein was carried out under the Naval Ship Research and Development Center (NSRDC) General Hydromechanics Research Program

Pro-ject SR 0090101 and the NSRDC in-house Independent Exploratory Development

Program Project ZFXX412001.

INTRODUCTION

Reference 1* treated the two-dimensional linearized hydrodynamic problem of a wedge penetrating a heavy water surface with a constant

velocity. The infinite Froude number approximation was found to be good for Froude numbers (VI j/E) of 3 or larger. The infinite Froude number approximation simplifies the boundary conditions considerably, i.e., a zero

acceleration potential on the free surface. This simplified model is used here for the more complicated problems dealt with in this report.

The water entry of a wedge or a foil with a finite chord may be

con-sidered in two stages: (1) the first stage between the time the leading

edges touch the water surface and the time the trailing edge enters the water and (2) the second stage after the trailing edge enters the water and

a trailing edge cavity is formed and entrained behind. Since the nature

of the problem is different for these two states, the solution to the

problem may be dealt with in two steps.

The present study used the method of conformal mapping to

and a base-ventilated symmetrical wedge. Only the first stage was treated

for the first two configurations because of the complexity involved. Only

the second stage was investigated for the symmetrical wedge since the

so-lution for the first stage had already been reported.1

Generally, the approach is to solve first the time derivatives of

the complex velocity w in the transformed half-plane for given boundary

conditions along the real axis. The conformal mapping is a function of the

coordinate of the physical plane and the time t. Unknown coefficients associated with w are determined by matching the boundary conditions of

the complex velocity w and the pressure. The pressure is obtained by

inte-grating w with respect to the physical coordinates.

The solutions of flow field, pressure, and drag are given in closed

forms for all the configurations treated here. This was possible because

of the following three facts:

The configurations all have straight sides which make the boundary

conditions for w all zeros in the transformed plane, unlike the cases of

cambered foils.

For the problems of the first-stage entry, the complex velocity w is

a function of time only through the time-dependent transformed coordinates;

for the problem of the second-stage entry, the symmetry of the problem

helps the simplification.

The numbers of juncture points are small enough for closed-form

inte-grals.

It was felt that this investigation would be extremely helpful in under-standing the physical behavior of flows due to impinging blades of partially

submerged propellers or to similar water entry with ventilation. Numerous linear theoretical treatments of water entry have been published,1 but a

literature search failed to find a linear treatment of water entry problems with ventilation. The investigation reported here was motivated by the potential application to partially submerged propellers for high-speed, high-power craft; such results will also contribute to an understanding of

Initially

FORMULATION OF PROBLEM

The free surface at time t < 0 is undisturbed and represented by

y = 0. The y-axis is vertically upward and the x-axis is on the undisturbed

free surface such that the conventional rectangular coordinate system 0-xy

is formed. Consider a two-dimensional foil with chord c

x = f(y + Vt) , O < y + Vt < c (1)

penetrating through the water surface vertically with speed V, touching the origin O at t = 0. Then there exists in the water the potential

x,y,t) which satisfies

+c

=0

(2)

xx _yy

at any time. On the free surface, if an infinite Froude number approxi-mation is used,

=

°'i

_'

= 0

within the linear approximation. From the Bernoulli equation, the

pertur-bation pressure p can be represented by

p=-ppt

(3)

Thus

= O (4)

on the cavity.

Within the linear approximation, this condition can be applied on the y-axis instead of the unknown cavity surface when the foil is thin.2

On the foil, the relative normal velocity is zero. Thus

X = V f (y + Vt)

on (5)

Q < y + Vt < c , x = O

= = O

At infinity,

decaying properly. Especially, for the water entry of a wedge1

Thus this is a mixed boundary value problem.

To solve this problem, Equation (2) is modified by differentiation

with respect to t From Equation (3) From Equation (4) From Equation (S) Let (7) = 2 1

2'

at 2 + y2

\X +y/

+q

=0

txx tyy tx = 0 on y = 0

ty = O on the cavity part of x = O

=v2

tx _yy

on

O<y+Vt<1,x=0

w(x,y,t)=

-iq

tx ty

Then the problem for w is exactly the same as that for the complex potenti

with the real and imaginary boundary conditions available. This kind of

problem can be solved by the method used by Cheng and Rott3 within unknown

constants. _{To obtain a complete solution, a conformal mapping has first}

to be set; then the analytic function w _{which satisfies its boundary} con-dition should be found; and finally, the unknown constants must be

determined to satisfy the original boundary conditions for w and =

-(7a)

(3 a)

(S a)

CONFORMAL MAPPING

At time t, the physical boundary configuration where the boundary conditions are applied is approximated

by straight segments A BCD A as

in Figures 1 and 2. By the Schwartz-Christoffel transformation,4 the

z = x + iy plane can be transformed to the ç = + in plane (Figure 3)

as

kç ₍₈₎

y/ç

-1

with the points B (z = + 0), D (z = 0), and C (z = -Vt i) mapped to ç = -1,

ç = 1, and ç = O respectively, where V indicates constant speed. The integration of Equation (1) leads to

1

z = K(ç2 1)2 + L (9)

Constants K and L will be determined by matching coordinates

r ç = 1, z = O _; hence L = O I (10) ç = O z = -Vti; hence K = -Vt Thus dz dç or

z = 2K/+ L

(12)

transforms the z-plane to the ç-plane with the correspondence:

çV2

= (Vt)2

z

(9a)

For problems of symmetrical wedges, only one-quarter of the physical plane may be needed as shown in Figure 4.

The mapping function

dz_

K

Thus it leads to

(14)

ç=O,

z=O

;

henceL=O

ç = 1, z = -Vti; hence K = -iVt/2 (13a)

ç = a, z = (_\Tt + l)i;

a - ívt - i

2

(l3b)

Also from Equations (12) and (13)

fz

\2

(:2a)

GENERAL TECHNIQUE FOR OBTAINING SOLUTIONS

The boundary conditions in the z-plane for w, which are analytic except at singular points on the boundary, do not change in the ç-plane on

the corresponding segments. The unique characteristic of the problems

concerned here is that the boundary conditions of w are all either real

parts or the imaginary parts of w. For the solution to this kind of problem in the ç-plane, a homogeneous solution must be found first; all

the given nonzero boundary values are replaced by zero, and the solution

is found for the same problem. For this, conditions at the juncture point must be specified. First it is reasonable to assume the Kutta condition4

at the trailing edge. At the leading edge there can be a certain

singu-larity as in the steady flow of a linear cavitating foil.5 At the joint

of the free surface and the cavity, there could be some singularity which

might indicate the possibility of spray phenomena. At infinity, the same behavior is assumed as that of the solutions for the wedge entry

w log

-2)

z - z

(1

z-z1

for a large

z.

Singularities of w and at the juncture points are

If the boundary values of w and juncture points in the c-plane are

not functions of t, w is a function of t only through r. Namely

wt w (15)

where = F (z, t) from the conformal transformation. Actually, from Equation (9)

2

- t

(16)

and from Equation (12)

2

-- t (17)

All problems treated in this report have homogeneous boundary

con-ditions for w

or w.

Thus the homogeneous solutions are our solutions.

They have a form of3

q1 q2 q3 _m

w=(+l)

(ç-1)

ç

(A+A.ç «.ç)

ç o

it

m

where q. is given one-half plus the integer or integers that can be used

to satisfy juncture conditions,

A are unknown coefficients that are functions of juncture points, and

m should be decided by the condition at .

TABLE i

Singularities of w and

(18)

w

Leading Edge of Wedge

Trailing Edge with Cavity Entrairunent

Juncture Point of Free Surface and Foil

At Infinity 1/ç l/ç2 log(ç-i) l/z2 l/(ç-l) l/z3 ----dç

The unknown coefficients are determined by matching boundary conditions of

w and the pressure; w is from

1.

ÇWt

w

= j

w d

= j

--dç

The pressure will be obtained from Equation (3) and

dz

= Re

J

w --- d

When the boundary condition on the foil is an arbitrary function of y + Vt,

set a function

H

- U

_-

i _VH

W u - i _v

(21)

where H is the homogeneous solution; then the imaginary part of wt/H will be known everywhere on the real axis of the -p1ane.3 Thus, the solution for w can be written

w

W

t

H 1Hd

1Tl (22)

-WATER ENTRY OF A NONSYMMETRICAL WEDGE

The problem of a symmetrical wedge has already been solved by using the Green function.1 The problem of the nonsymmetrical wedge will be a

little more general. Consider that a wedge whose two side planes respectively form angles a1 and a2 with the y-axis enters the water vertically with speed V. Then the boundary conditions on the wedge are

Thus

j-a2inü<y+Vt

x=-O

f (y + Vt) = y

a1inü<yVt x=+0

f),t(>T +

Vt) = fyy = O in O _< + Vt x 0 (24) (23)

Hence a possible solution for w is obtainable from Equations (9) and (14) through (16) and the juncture conditions shown in Table 1

= 2 2 -

2(+l)

2(-l)

A +B

R

-1) B

_{- -+}

A

A-Bk

_A+BI

(25)

where A and B are unknown real coefficients to be determined by matching

conditions. From Equation (25) w =

J

w d i(B-A) /i . A+B Il

+lo

+1

-

2 g l+ç) 2 log

--)

(26)

From the boundary conditions of Equations (23) and (25)

B- A

2

A+ B

2

--a2V/T

or = Re V V

A- (a1 + a2), B= (a

-

a)-1

2i

The pressure will be obtained from Equations (3) and (20)

1 1 dz dz = - _{= Re}

J

w -- d = Re

J

w d p t ç

-.2

V

Tr

f(a1+

a2)ç+a2

-2

_ç

ç

-

I (a - a2)

ç (a2 + a1) log

2

Tr 1

1

+Vlç2

(27)

when a2 a1 = a p 4 _ç 4 l/f'Vt)2_ _y2 = log = log

-

1+

_yÇi?

-:;:-

_vt+

y

aV2

2 log - Vt

--

ono>y> -Vt

(29) y + Vt

which was obtained previously.1

Thus the first term of the right-hand side of Equation (28) is due

to the nonsymnietry of the wedge. From Equation (26) this is clearly represented by a vorticity at ç=O with the strength proportional to

in the ç-plane.

This problem can also be easily solved for the complex velocity w = u - iv instead of w, since u is available everywhere on the boundary

of both the free surface and the wedge. The result is exactly the same as

that obtained here.

The drag coefficient of a nonsymmetrical wedge is from the

inte-WATER ENTRY OF A FLAT PLATE WITH LEADING EDGE CAVITY

Consider that a flat plate is entering the water surface vertically with a high speed V at an angle of attack a so that a cavity is formed

from the leading edge. When the flat plate has a finite chord c, the time

for the foil to fully enter the water is approximately c/(Vt). However, only the stage before full entrance will be considered here.

Since the pressure on the cavity is zero, the boundary values for w

gration of Equation (28)

D

_2

2

-

(a1 + a2)

log 2 +

(a2

2J

- a1)

p V Vt

The lift coefficient is

L 4

la

-ir I 2

a

_l

1 2 Vt

homogeneous. From Equations (18), (19), (14), and (15) as well as the juncture conditions in Table 1, the solutions for w can be written as

i (A+A1

w =

ç

ç(ç-l) (ç+l) )t'ç(ç+l)

Coefficient A will be obtained from the boundary conditions of w which

can be obtained by integration of wç

C w

11

i w dç i i J ç 3ç i + + j/ç(ç-i-l)

2A

A+A

+1 o 1

2'

2/

--

10

2'

ç - 1

A +A

o 1

3+2"

+ (A_A)J/Ç+2A +

log o 1 °

2)7k

2/

- (A0-A1) (31)

The flow condition on the pressure side of the flat plate O < ç < i and the

quality of the pole at ç = i give

A + A 2)1v

o 1

For the condition on the cavity and the free surface, is obtained from Equations (3), (20), and (30)

i=-

=

_Ref

w

dz p t

tdç

ç ' ki (A0+A1ç) Re tç(ç+1) ç(ç-l) dç (30)

thus

A

= A ---aV

i o r

Therefore the pressure on the foil can be written

rp

-

Re

2//i

a p

The drag is 0 1

Vap

22

Vap

2 rrD p dy = Im 2

yT

Ç d -Vt 6 1

=im2iìJ

K/

_{d=2V'Vt JyÇÇdc}

o = 2 }/Y Vt ¡

+l) -

log ( + + i D - _{J. {8 - 2}

_3/

_{log (3 + 2} = 0.995 1

22

VapVt

Tr

Re['I2

A-A

-

A log o + i -1 1 o

2'

2/

-3 1 + i A1-A0 3

iog(

- l)Ñ

(33) t I

°

\2V'

(Cont'd)

To have p satisfy boundary conditions on the free surface jj>i and the

cavity 0>r>-1, it is necessary that

A -A =0

i o

(3

The approximate cavity shape of the ventilating flat plate can be obtained from y x = Re

J

w dy =

J

u()

d =- Vt

_{f u(a)}

d -Vt O O O > > -1 using Equation (8).

From Equations (31) and (35)

aV (2

VT'-

+ -1 2

j"

J/(l-)

Re w = u = -

TI

i - 3

TI

Thus the above integration leads to

z

2V(l-)

> ¡2

(l+)

i (2)

tan 1 - 3 Vt TI

-

y/(Vt)2

O > y > -Vt

BASE-VENTED SYMMETRICAL WEDGE

Up to this point, all the problems considered herein are related to the first stage before the full entrance of a wedge (or foil) of finite

chord.

After the full entrance, ventilation will be entrained from the trailing edge and sometimes from the leading edge. A water entry problem of this kind is extremely complicated for a general foil. However, when

the case is a symmetrical wedge entering the water surface vertically with

speed V, the problem becomes fairly simple.

1}

X

Vt

For the pressure

=

(c-l)

(-a)

where A and A1 are not functions of but possibly functions of t. Sub stituting =- z2/(Vt)2 from Equation (12a)

2 \) t

ti'A

+A

Z 2)Idt w =

J

w dt =

J

0

(

(Vt) 2 2 (39) o

°

(

(Vt)2 1) (Vt)2 ₍ (Vt)2 - a)

Here the integral is imaginary when Vt

_<Iy

at a given point on the y-axis. For -y + i > Vt > -y,

Rew=cLV

(4 )

This value is observed to be from the singularity at y =- Vt in the inte

grand of Equation (39). Thus

2V

Tr (A + A1) Vt dz

=Re

J

wt-a_d

A +A

ç Vt ° (3

=cV

(4

In this case, the flow is symnietrical with respect to the wedge

axis. Therefore, only one-quarter of che physical plane or the z-plane

can be conveniently considered. The conformal mapping to the ç-plane is

already considered in Equations (il) through (13). The boundary conditions

of w are again homogeneous. Considering the behavior of singularities at

juncture points, including infinity, with Equations (7a), (11), (12), and (20), it is seen that w can be written

i(A

+ A1)

8)

p Vt p 2 A

2A

A +A

_{o i} o -1 a tan log lo

-A +-A

° log -

vT

Vt

ÏT

y+

V12

To have the pressure satisfy the boundary condition on the free surface

A =0

o

Therefore,

2 o.

V2 yii

Vtr

Thus the pressure on the wedge surface is given

=---lo

;ìii+

/ii

+V2p

which satisfies all the boundary conditions.

When Vt = 1, a = O from Equation (13b). Then

p

- log

12

p

with =- z2,(vt This is already given in Equation (29).

For Vt - =), _{z x + (-Vt + y1)I, >1} 2o.

1-y1+l

= - log

V2p

(42) (46) (47)

which is the case of a steady cavitating wedge in an infinite medium

having a zero cavitation number.

The drag due to the wedge from the stage Vt/c = 1 will be - Vt + 1

D=2aJpdy

Thus where a =

22

2cL V p = Im log

i

J

--iog

=

Vt

hm

EO

=

um Vt

-

log log

V+ VT

y-

yiT

(i

-

-

-;-+ V(-a)(l-a

(c-l)

/i

D 2 2 p

-

[log a +

2-a2

]vt

log a ¿

l-E

l-E

--viJ(l-c)

p'i1

a a

I

l-E

a

Vt

l\ )

from Equation (13h). For a = O or Vt = 1.

D 4 = _{log 4}

22

C (48)

(49)

V+

d

V-

yiT

log

V-Vî

For Vt - either from Equation (48) or from the integration of Equation

(47),

V2a2

=

(50)

which is the drag due to a two-dimensional cavitating wedge with the

cavi-tation number equal to zero2 in an infinite medium.

DISCUSSION

The pressure distributions on both sides of a nonsymmetrical wedge entering water for the wedge angles a1 + a = 0.2 and 0.1 radian are shown

in Figures 4a and Sa, and 4b and Sb, respectively, for various values of

- which are twice the angle between the y-axis and the centerline of the wedge. The effect of asymmetry comes from the first term on the

right-hand side of Equation (28), which is positive on the pressure side

of the wedge and negative on the suction side. The singularities of both the complex velocity and the pressure near the leading edge due to the asymmetry are like l/}/l-y/(Vt), while those for the symmetrical wedge are

logarithmic. Therefore, in the immediate vicinity of the leading edge,

this linearized theory has to be interpreted differently as in the case of

usual linearized-aerofoil theory.

Figure 6 shows the pressure distribution on the pressure side of a nonsymmetrical wedge for a certain angle a between the pressure side of the wedge and the y-axis, with varying angle ra (r < 1) between the suction

side of the wedge and the y-axis. It is interesting to see that when r decreases, the pressure increases very little on the pressure side.

Eventually, the suction side will cavitate at some small r like a flat

plate. Also note that there is very little difference between the pressure distributions on a cavitating flat plate and a symmetrical wedge, as shown

in Figure 7.

The leading edge singularities of the complex velocity and the

pressure for the ventilating flat plate entering water are like

(l-y/VtY4

The drag coefficient D/(1/2 p V2 Vt) and the lift coefficient L/(l/2 p V2 Vt) for a ventilating flat plate entering water are compared in Figure 8 with those of a nonsymmetrical wedge entering water for the

wedge angles a1 + a2 = 0.2 and 0.1 rädian. The drag coefficient of a

symmetrical wedge, i.e., when ci1 = a2 = a, is 1.75 a2 as shown in Equation

(49). But only one-half of this drag is due to a single side of the wedge.

When this is compared with the drag coefficient of the cavitating flat plate, the ratio of the latter to the former is 1.09, showing little

con-tribution from the difference in leading-edge singularities.

The change of the pressure distribution on base-vented symmetrical

wedges after full entry is shown in Figure 9. The increase of the pressure

22

rapidly approaches the steady state. The drag coefficient D/(l/2 p V a c)

of the base-vented symmetrical wedge is shown in Figure 10; the maximum drag increase of approximately 45 percent occurred for the period Vt

= C

from the time that the trailing edge was instantly on the mean free

sur-face to the time when steady state was reached.

Under the present assumption that the cavity is from the free sur-face to the foil without any cavity closure or sursur-face sealing, there appeared to be no singularity for either w or p at the juncture point of

the free surface and the cavity. However, at the juncture point of the foil and the free surface, there was a logarithmic singularity for w.'

All the cases concerned here are particularly simple especially in

the c-plane. However, the results permit more physical information to be

easily extrapolated to aid in understanding actual complicated situations. Flow fields for a nonsymnietrical wedge and a cavitating flat plate can be obtained directly from Equations (26) and (31), respectively. For example,

the cavity shape due to the ventilating flat plate entering the water is

shown in Figure 11. Note that the cavity thickness is much smaller than the steady state case of a cavitating flat plate beneath the free surface.6 The behavior of the pressure field after the full entry of a finite

cavi-tating flat plate can be estimated from the case of the finite symmetrical

The solutions obtained thus far are for the problems of water

entry. A reversed application of the case of entry is possible for an

exiting foil that is not cavitating.7 However, the problem is much more complicated for an exiting, cavitating foil. This problem will be con-sidered in the near future.

ACKNOWLEDGMENTS

The author thanks Mr. J.B. Hadler, Dr. P.C. Pien, and Mr. H.M. Cheng for their interest and the valuable suggestions which contributed

A A -A (c)

txo

D = -a2V yi (b) E C

-

IC = a1 B C O

Figure 1 - Nonsymmetrical Wedg

A

A cO

A

(b) y e D

4x=-av

(c) o

A

4tx

B C D

Figure 2 - Ventilating Flat Plate

B _A

0

Y A co

-o

Dco c)

Î

B E C Doe

Figure 3 - Base-Vented Symmetrical P B X

Ax

t_o

(-Vt+I)

av

- Vt i X O E C

0.8 O

-0.2

-0.4 -0.6

-08

-1.0 y/vt

Figure 4a - Wedge Angle

+

= 0.2 Radian

0.2 _0.15 0.1 _0.05

a2 --a

0.1 rad

Figure 4 - Pressure Distribution on the

Pressure

Side of a Nonsymmetrical Wedge Entering Water

'a -a

=0.02

0.04

-LO

y _Vt

Figure 4b - Wedge Angle c1

+ c2

= 0.1 Radian

0.06J

L... -0.2 -0.4 -0.6 -0.8 P V2

P - V2 0.30

005

+ a1 0.2 RAD 0 -0.2 -0.4 -0.6 -0.B -1.0

y/vf

Figure 5a - Wedge Angle c + = 0.2 Radian

=0.1 rad

0.2

0.4

Figure Sb - Wedge Angle + a2 0.1 adiafl

Figure 5 - Pressure Distribution on the Suction

0.6 y Vt 0.05 P pV2 0 0.05

P

aV2 3

2 o

-02

(SYMMETRICAL WEDGE) -0.8 -IO y/vt

Figure 6 - Effect of Changing Angle on Pressure Distribution on the Pressure

Side of a Nonsymmetrical Wedge

VENTED FLAT PLATE

SYMMETRICAL WEDGE

I

I

Figure 7 - Pressure Distribution on a Vented

Flat Plate Entering a Water Surface

o -0.2 -0.4 P 2 -0.6 -0.8 -1.0 yi Vt

D

C-

D1

V2Vt

2 L CL = _pV2Vt , ,

/

,

a12=O.2ra

/

,

/ /

T

/ /

I,

NON SY MME TR IC AL WEDGE 0L FØ2 rod V F4T IL AT ING

/

FLAT PLATE

/

I I I 0.04 008 012 0.16 0.2 a, a2 - ø I rod

Figure 8 - Drag and Lift

Coefficients of

Nonsymmetrical Wedges and

Ventilated

Flat Plates Entering Water

3. o -0.2 -0.4

îvet

-0.6

-08

-.

X y,C+VtJC-I

Figure 9 - Pressure Distribution on

Base-Vented

Thin Wedge Entering the Water at Fr

- Lo

6. 4. 2. 1,-P _apV2

3.0 2.5-D calVIP 2.0 2 '.5 -0.8 I I 2 3 4

Vt/c

Vt 2a

Figure 10 - Drag Coefficient of a Base-Vented Symmetrical Wedge Entering the Water with

Ventilation y Vt -1.0 -0.8 -06 -0.4 -0.2 0 0.2 04 0.6 0.8 1.0

/

/

/

/

--0.2

II

/

/

/

--0.4

/

/

/

/

/

--0.6

/

/

/

/

/

/

/

-1.0

Figure 11 - Cavity Shape of a Ventilated

Flat Plate Entering Water

X

Vt.

2.0 irD ZaZCPV! I .5 1.0

REFERENCES

Yim, B., "An Application of Linearized Theory to Water Entry and Water Exit Problem Part i (with Cavity Effect)," NSRDC Report 3041 _May

1969)

Tulin, M.P.., "Steady Two-Dirnen:ional Cavity Flows about Slender

BodiLs" David Taylor Mdel Basin Report 8.4 (May 1953).

. Cheng, H.K. and Rott, N., "Generalization of the Inversion

Formula of Thin Airfoil Theory," Journal of Rational Mechanics and

Analysis, Vol. 3, No. 3 (May 1954).

Milne-Thomson, L.M., "Theoretical Hydrodynamics," Fifth Edition,

The MacMillan Company, New York (1968).

Tulin, M.P., "Supercavitating Flows _{- Small Perturbation Theory,"}

Journal of Ship Research, Vol. 7, p.o. 3 (Jan 1964).

Yim, B., "On a Fully Cavitating Two-Dimensional Flat Plate Hydrofoil with Non-Zero Cavitation Number Near a Free Surface," Hydro-nautics Incorporated Technical Report 463-4 (Jun 1964).

Moran, J.P., "The Vertical Water-Exit and Entry of Slender Symmetric Bodies," Journal of the Aerospace Science, Vol. 28, No. 10

1 Willi'm H. Mueller,

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UNCLASSIFIED

Security Classification

DOCUMENT CONTROL DATA - R & D

Security classification of title, body of abstract arid indexing annotation niust be entered when tine overall report is classified) ORIGINATING ACTIVITY (Corporate author)

Naval Ship Research E1 Development Center

Washington, D.C. 20034

20. REPORT SECURITY CLASSIPICATION

UNCLASSIFIED 2b. CROUP

3 REPORT TITLE

AN APPLICATION OF LINEARIZED THEORY TO WATER ENTRY AND WATER EXIT PROBLEM

PART 2 (WITH VENTILATION)

4 DESCRIPTIVE NOTES (Type of report and inclusive dates) S AU THORISI (Fitsl natrne, middle initial, last name)

Bohyun Yim

B REPORT DATE

July 1970

70. TOTAL NO. OF PACES

33

7h. NO. OF REPS Ba. CONTRACT OR GRANT NO.

b. PROJECT NO SR 009 0101

s. d.

98. ORIGINATORS REPORT NUMBER(S)

3171

Sb. OTI-IER REPORT NO(S) (Any other numbers that may be assigned

this report)

5 DISTRIBUTION STATEMENT

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

is unlimited.

II. SUPPLEMENTARY NOTES 12 SPONSORING MILITARY ACTIVITY

Naval Ship Research E1 Development Center

General Hydromechanics Research Program

13 ABSTRACT

A conformal mapping technique is used to analyze two-dimensional thin foils entering a water surface vertically with ventilations. Boundary conditions are linearized, and

the gravity effect is neglected. Flow fields, pressure

dis-tributions, and drags are given in closed forms for an in-finite nonsymmetrical wedge, an inin-finite flat plate with ventilation from the leading edge, and a finite base-vented

UNCLASSTFT PD Security Classification

FORM 4473

4

KEY WORDS LINR A LINK B LINK C

ROLE w-r ROLE W ROLE WT

Water entry of flat plates Ventilation

Nonsymmetrical wedge

Pressure

D rag