On the interaction between a strut and the free surface

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No. 175

August 1975

ON THE INTERACTION BETWEEN

A STRUT AND THE FREE SURFACE

T. Francis Ogilvie

This research was carried out under the Naval Sea Systems Command General Hydromechanics Research Program, Subproject

SR 023 01 01, administered by the David W. Taylor Naval

Ship R & D Center Contract No. N00014-67-A-0181-0053

Reproduction in whole or in part permitted for any purpose

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Approved for public release; distribution unlimited,

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_OF

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410ot-clot-AND

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4061-THE UNIVERSITY OF MICHIGAN

COLLEGE OF ENGINEERING

-Bibliotheek van de

Afderag Sc4cep,s',:euw-cn Schecpynartunde

Technisce Ho,-,esch7.0!, Dept

DOCUME;STATiE :

9_ 4-15

DATUM'

ON THE INTERACTION BETWEEN A

STRUT

AND THE FREE SURFACE

T. Francis Ogilvie

This research was carried out under the Naval Sea Systems Command

General Hydromechanics Research Program, Subproject SR 023 01 01, administered by the

David W. Taylor Naval Ship R&D Center Contract No. N00014-67-A-0181-0053 Reproduction in whole or in part permitted for any purpose of the United States Government. Approved for public release; distribution unlimited.

1,,,V1Y0,,,. _{Department of Naval Architecture}

44,

11V li

a. and Marine Engineering

.... _ci

s

College of Engineering

.o

.0 op The University of Michigan

7111 Ann Arbor, Michigan

48104 No. 175 August 1975

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4. TITLE (and Subtitle)

ON THE INTERACTION BETWEEN A STRUT

AND THE

FREE SURFACE

5 TYPE OF REPORT & PERIOD COVERED

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T. Francis Ogilvie

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Sponsored by the Naval Sea Systems Command, General Hydromechanics

Research Program administered by the David W. Taylor Naval Ship Research

and Development Center, Code 1505, Bethesda, MD. 20084

19 KEY WORDS (Continue on reverse si,je if neces,ary and identify by block number

STRUT

SURFACE-PIERCING STRUT

SLENDER-BODY THEORY

BOW WAVE

GHR Program

20. ABSTRACT (Contint, on reverse side If necessary nnd identify by block number)

A procedure is proposed for solving problems involving the

flow around surface-piercing struts.

There are three steps:

(1) An infinite-fluid problem must be solved, in which the flow

is antisymmetrical with respect to the plane of the undisturbed

free surface.

(2) From that solution, there is effectively a

distribution of normal velocity imposed on the plane of

anti-symmetry; the resulting wavelike disturbance can be found by a

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procedure which is effectively a slender-body analysis.

(3)

The wavelike disturbance produces a disturbance

on the body

(strut), which must be canceled by yet another wavelike motion,

which can also be found locally from a slender-body analysis.

In order to justify the second and third steps, some

difficul-ties and advantages associated with slender-ship theory

are

examined.

It is shown how the usual axial source distribution

of slender-ship theory can be modified systematically

_{so that}

the slender-ship wave resistance becomes identical to the

classical result of Michell for a thin ship.

The slender-ship

procedure has the further advantage that it is not limited to

thin bodies; in particular, the body boundary condition should

logically be satisfied on the exact location of the body surface,

ABSTRACT

A procedure is proposed for solving problems involving the flow around

surface-piercing struts. There are three steps: (1) An infinite-fluid problem

must be solved, in which the flow is antisymmetrical with respect to the plane

of the undisturbed free surface. (2) From that solution, there is effectively

a distribution of normal velocity imposed on the plane of antisymmetry; the

resulting wavelike disturbance can be found by a procedure which is effectively

a slender-body analysis. (3) The wavelike disturbance produces a disturbance

on the body (strut), which must be canceled by yet another wavelike motion,

which can also be found locally from a slender-body analysis. In order to

justify the second and third steps, some difficulties and advantages associated

with slender-ship theory are examined. It is shown how the usual axial source

distribution of slender-ship theory can be modified systematically so that the slender-ship wave resistance becomes identical to the classical result of Michell

for a thin ship. The slender-ship procedure has the further advantage that it

is not limited to thin bodies; in particular, the body boundary condition should logically be satisfied on the exact location of the body surface, not on the centerplane.

CONTENTS

INTRODUCTION

FAILURE OF SLENDER-BODY THEORY AT THE BOW

ADVANTAGES OF USING SLENDER-SHIP THEORY 13 ,

METHOD FOR SOLVING STRUT _PROBLEMS 18

REMAINING DIFFICULTIES 22

REFERENCES 25

APPENDIX /2 EQUIVALENCE BETWEEN TRANSLATING 31) SOURCE, AND

IMPULSIVE 2-D SOURCE 26

1

INTRODUCTION

This study was started because of the observation that the wave created at the bow of a fine wedge can be predicted fairly well (Ogilvie (1972)) on

the basis of a slender-body theory. In that analysis, it was assumed that an

incident stream experiences absolutely no disturbance upstream of the fine entrance of a slender body; this is not true, of course, but the assumption

nevertheless leads to predictions that are fairly accurate. The analysis was

not based on the conventional assumptions of slender-body theory, but it was

nevertheless a slender-body theory and as such it may be expected to be less

valid if applied to a strut-like body, although the example worked out indicated that this difficulty might not be so serious as intuition suggests.

The purpose of this study is to apply a similar concept to problems of surface-piercing struts, to determine the limitations of such an approach, and to develop modified procedures as far as possible to overcome those limitations. A major side-result has appeared too, and it may be much more significant than

the original goal: It is found that the problem can be broken into two parts,

one of which is a conventional infinite-fluid problem, the other a problem in wave propagation which can be treated in much the same manner as problems in which the governing equation is a typical hyperbolic wave equation.

In the light of classical ship hydrodynamics, it might be wondered why anyone would even suggest applying slender-body theory to a strut problem;

thin-ship theory might be a more reasonable approach. Aside from the

demon-strated partial success of the earlier analysis, the following reasons may be cited for at least considering the slender-body approach:

1) Thin-ship theory is based on the assumption that the field variables

change slowly in the lateral direction near a ship. This is actually true over

most of the surface of a ship, but it is wrong near a ship bow. A strut may be

visualized for some purposes as a fine ship bow (without the rest of the ship!),

and so the thin-body approach may be expected to fail in a strut problem. As

will be seen, slender-ship theory does not necessarily suffer from a correspond-ing handicap.

1 2 1

-Directly behind a surface-piercing strut moving at an angle of attack,

there is a discontinuity in free-surface elevation; behind a symmetrical strut,

there is a discontinuity of slope of the free surface. Such discontinuities

can be analyzed on the basis of a slender-body approach, but no valid trea-ment

based on a thin-ship model has been found possible. This problem is potentially

important in determining the effect of a forward strut on after struts. Within the framework of thin-ship theory, the solution of problems

involving lateral assymetry is terribly difficult. The computational problems

can readily be seen in the work of Daoud (1973), who solved the case of a flat

plate of zero thickness moving ahead with a steady yaw angle; the

two-dimen-sional singular integral equation of linear lifting-surface theory becomes

almost intractable because of the inclusion of free-surface effects. Such problems can be much less complicated when treated within the framework of

slender-body theory.

Of course, there are some serious objections to applying slender-body

theory to strut problems. Ordinary slender-ship theory, as developed for

example by Tuck (1963), fails catastrophically when applied to ships with

wedgelike bows, no matter how fine they are. As one manifestation of this

failure, the theory predicts infinite wave resistance for such a ship form;

this ridiculous result is the result of errors in the theory

near the bow,

and so one might expect a similar failure in the treatment of strut problems.

In addition, of course, one must face up squarely to the objection that

a strut is not a slender body, and any attempt to analyze strut problems by

slender-body theory requires considerable justification.

In this report, the failure of the slender-ship theory is examined for

the case of a ship with a fine wedgelike bow. It is shown how slender-body

theory can be modified so that the far-field expansion (to one term) is identi-cal to that of thin-ship theory (which is essentially a far-field theory). This equivalence shows that the bow anomaly in slender-ship theory can be avoided.

In developing the above equivalence, we find that the finite draft of the

ship, particularly near the bow, causes the generation of waves which seem to

arise from points upstream of the bow. The significance of this result is not

3

-entirely understood. It may possibly explain the observation reported from

time to time that ship waves seem to occupy a Kelvin angle with apex forward

of the bow. However, some reservations may be in order with respect to this

interpretation.

In any case, it seems evident that a strut-like body must cause more upstream disturbance than a slender body of shallow draft, and such upstream effects will cause havoc in a slender-body analysis predicated on the

non-existence of upstream disturbances. A method for treating such effects is

proposed. This method does not appear to offer the possibility yet of being

applied in a routine manner, but, if some special cases can be computed, they may indicate the extent of validity of simpler procedures.

It may be noted that the proposed method shows explicitly that the origin of ship-generated waves ought to appear partly to be ahead of the ship.

The general plan of attack here is to resolve first those special diffi-culties that might prevent the application of slender-body theory to strut problems and then to develop that application so as to take advantage of the special advantages of slender-body theory.

FAILURE OF SLENDER -BODY THEORY AT THE BOW

It has long been recognized that slender-body theory gives poor predictions of wave resistance, and this shortcoming has been blamed on the fact that, in

the far-field representation, the slender-body source distribution lies entirely

on the level of the free surface. Thus, the strong decay with depth in

wave-making effectiveness of submerged sources does not appear in the slender-ship representation.

This interpretation is misleading, but it is nevertheless instructive to

0

is the ship centerplane,

0

b(x,z) is the hull offset at (x,z) = g/U2 .

-4-recall how it comes about. One form of Michell's integral for wave resistance

is the following: R 40a 2

"

=

I

7/2

₃₆ [p2(6) 4. Q2(6)] de sec (1) TrU2 where 0 P(0) +

i

oe)

=

ff

b_x(x z) eKzsec2eeiKxsece dx dz (2)

In the Michell-Havelock theory, bx(x,z) is proportional to the local source

density, a(x,z) , in fact, a(x,z) = 2Ubx(x,z) . If all of the sources are

concentrated at z = 0 , and if we define

0

E(x) = 2U dz bx(x,z) = U A' (x) , (4)

-H(x)

where H(x) is the keel depth at station x and A(x) is the area of the

submerged cross-section, then

1iKx sec

eE

(x)

P + iQ = d

2U x e = 2U

E* (K sec e )

(5)

where E* is the Fourier transform of E(x) and L is the domain of the ship

length. Michell's integral then becomes:

K3)

1

5

which is a standard form for the wave resistance of a slender ship, as given,

for example, by Tuck (1963); Yo is a Bessel function in the notation of

Abramowitz and Stegun (1964). The source density, E(x) , is defined to be

identically equal to zero outside of the ship.

This correspondance between thin-ship theory and slender-ship theory has been noted by numerous writers, most of whom have then assumed that the poor wave-resistance predictions of slender-body theory are caused by setting

z = 0 in the integrand of the expression for P+iQ, that is, in (2).

However, for a wedgelike body, the failure of the slender-body formula

When this is substituted into (7), the answer is indeterminate (infinite). The

same is true if we compute E*(k) for substitution into (6): In the sense of

generalized functions, E*(k) contains one term, -2UHai/k , which leads to a

divergent integral in (6).

Michell's integral gives a reasonable prediction (even if not very accurate)

is catastrophic. Let the

being at x = 0 . If the

ship be located between x = 0

half-angle of the entrance is

o x < 0 and x = L , a , then the bow A (x) = (8a)

in the neighborhood of the

2Hotx + 0 (x2) , 0 <

bow. It follows that

x , E (x) = UA' (x) =

(0

x < , (8b) E'(x) = 2UHa 2UHa + 0 (x) d(x) + ,

0< x

(8c ) pK R = 2 7r/2 2

de sec 3e IE*(K sec

HI

0

k2

dk

1E*(k)12

(6) 7 P _f -Ik2 -K2 K

The last form can be changed into the following:

R =

-LId

dx E'( ) E' (x) Y0(Klx-) (7)

2

0

6

-for the wave resistance of a body with a wedge bow, and so the error in

slender-ship theory involves much more than just the inaccuracy of overlooking

the depth attenuation of waves generated by submerged sources. The

disconti-nuity at the bow end of the line of sources in the slender-body far-field repre-sentation is completely unacceptable for certain purposes -one of which is the

computation of wave resistance. Such a discontinuity is intolerable because

it leaves behind in its track an entire line of singularities. The waves so

generated carry out energy at an infinite rate. If the line of sources were

submerged the slightest amount, this would not happen. Thus, allowing for

depth of source submergence (at least near the body ends) changes the entire character of the free-surface flow.

The wavemaking properties of a submerged source distribution can be represented by a longitudinal distribution of sources along the free surface

above the location of the actual sources. When this has been demonstrated, it

will be apparent how to fix up slender-ship theory for the region near the bow.

One form of the Kelvin-Havelock unit-source potential is the following:

1(1

1 ]

cPs(x,Y,z)

74,7i _r'

v(

412 + k2eim(z+c) 472 1

I

dk eikx dm m e-l-m 1 I iK2 -k2 dk k cos kx e K e-kly1/1 - k2/k2 71. 0 -1 1 dk k

i/k2K2

e- LK2i z+ COS kx sin (Id y I Vk2/K 2 -1)

k21,7_,

1K/k2 K2

r 1

1 dk k

e- K 1 - 'I sin k_x cos (kyVk2/K2 - 1) .

7

-This is very similar to the form preferred by Michell himself; it can be derived by using separation of variables on the Laplace equation, then superposing all solutions obtained in that way so that the proper singularity is represented and

the free-surface condition is satisfied. In (9), the source is located at

(9) im2 k2 _{m + ik2/K} =

co

-K .

7

-(0,0,C) , with c < 0 , the stream of speed U flowing in the positive x

direction; also, r = [x2 +y2 + ) 2] 1/2 r = [x2 +y2 + (z+) 2] 1/2

The last term in (9) is odd with respect to x , and the other terms are

all even. The odd and even terms separately represent wavelike motion both

upstream and downstream of the source. The upstream waves must cancel, and so

there is a doubling downstream. Accordingly, we can represent all wavelike

motion downstream of the source by taking twice the odd term in (9), i.e.,

2 dk k k2,

e K sin kx cos (Jsyik2/K2- 1) for x > 0 .

(1)S(x'Y'z) K2

(10)

We can go yet one step further to obtain a more useful expression: Rewrite

the integrand using the relationship: 2 sin kx cos (kp/k2/K2 -1)

k 42/K2 -1)

sin (kx + sin (kx +kly1 )k2/K2 -1) .

Downstream of the source, only the first of these terms gives a significant wave motion, since the second has no point of stationary phase. Therefore we can write approximately:

k21

1 rdk k

- --iz+C

q)s(x,y,z)

=

-2

K'

e < sin (kx - /k2/K2 - 1) . 7

i/k-It is no longer necessary to add the restriction that x > 0 , since the above

integrand has no points of stationary phase for x < 0 .

The potential for the wave motion generated by a thin ship can now be expressed approximately as follows:

q)Ts(x,Y,z) = - -₇ dE

OC a(E,)

1 IL f°

fdk k

e-tlz

+c!

_s

.

in[k(x-E) - kI y116c2/K2-1]

0 -H (E) K ₍₁₂₎

where a(x,z) = 2Ubx(x,z) on the centerplane, as before. Extend this definition

by requiring that a(x,z) E 0 outside the centerplane. Now define:

0

/K p*(k) = dx e-ikx dz a(x,z) ek2z

-H(x) We then have:

(13)

-(PTS(X,Y,Z)

1 K

r-dk k

J] ]

A2

-K2 ek2z/Keik(x-Iyil/k2-K2/K)p*(k) . (14) co K

On the other hand, the wave motion generated by a line of sources of density E(x) along the axis is given approximately by the potential:

K

1

LI

dk k ek2z/Keik(x - lylik2-K2/K)z.(k)

4)55(x,Y'z) - 27i _{ik2 -K2}

These are the same if we identify E* and p* , that is, if

E*(k)=

j dx e-ikxE(x)

0

= dx e-ikx dz a(x,z) ek2z/K (16)

-H(x)

The significance is shown in some special cases. It should be noted, in

particular, that (14) and (15) represent the same wave motion if (16) is satis-fied. A consequence is that the two lead to identical values of wave resistance.

Point Source. Suppose that a(x,z) = a06(x)5(z-z0) . Then:

E*(k) = a0ek2z0/K

which implies that

.,/K

a

E(x) . -21

f

dk eikx ek2z u

27r -. / K KX = Cj01 47IZ0 e 0 ( Zo < 0 ) .

The total strength of E , that is,

f

E(x)dx , is the same as the strength,

a , of the original source, but it is spread out over the longitudinal axis.

0

In principle, it is spread out over an infinite length, but actually it is

fairly well concentrated, especially if the source is shallow.

In no sense does the line distribution produce local effects comparable

to those of the original point source. In fact, the potential corresponding

to the line distribution is analytic in the lower half-space, even at the

loca-tion of the "equivalent" point source. What is important for our purposes is

(15)

(17)

9

-that a free-surface line distribution of sources can produce the same waves as the submerged point source, but the line distribution extends fore and aft of

the point source. Furthermore, the distribution is analytic in x .

Vertical Line of Sources. Suppose that

a06(x) , -H < z < 0 0 , z < -H . Then: Elc(k) = L.S)"_ (1 e-k2H/K k2 and: 1 I

Kx2/411

E (x) = 00{- 'XI - erf /Kx2/4H} +

ViTrr

KI- e (18')

A longitudinal distribution of sources given by (18') produces the same wave motion as the vertical line of sources of constant density a per unit height.

0

It may be noted that E(x) is continuous, but its first derivative is not.

It is convenient to define a function which is proportional to the above E(x)

-KX

K!xlr 2/4H

M(x) =

2H (1 erf VKx214H) + e

It may be noted that FriT<M(x)

is

dimensionless and depends on the single

variable, X = /<x2/4H , that is,

1 -X2

VH/K M(x) = - X [1 - erf X] + e

)77

This function is displayed in Figure 1. It is a kind of spreading function.

Strut-Like Body. Let a symmetrical strut be defined in terms of its

offsets as follows:

y = ± b(x) , 0 < x < L , -H < z < 0 .

The corresponding thin-ship source distribution is given by:

a (x,z) = 2 U b' (x) ,

0< x < L

, -H < z < 0

(19')

,

(18)

M* (k) =

10

-I x-I

FIGURE 1.

THE FUNCTION

A

M(x) EXPRESSED IN .TERMS OF THE SINGLE VARIABLE Kx2

Ix!.

4H

From the previous result (for a vertical line of sources), we can immediately write down the corresponding longitudinal source distribution:

E(x) = 2UH dE b' (E)

M(x-E)

.

(20)

0

Ordinary slender-body theory gives the same result if M(x)

is

replaced by

(5(x) , that is E(x) = 2UHb'(x) . The representation in (20) shows that the

ordinary slender-body-theory source distribution is smeared out.

Since (20) has the form of a convolution integral, we can readily rewrite it in terms of Fourier transforms:

1

E(x) = dk eikx M*(k)

E(k)

27 where

Kr

0_ e-k2H/K) J k'H (21) (21')

the transform of (19), and E*0(k) is the transform of the source density

func-tion in ordinary slender-body theory, that is, in the strut case,

0

to

_1.2

1.4 1.6 1.8 2.0

.2

-T (X) =

specifying the keel line). We can then find E(x) explicitly, as follows,

aUK f dk eikx[l

-e-k2H/1{1

- 2e

E(x) = -iLk/2 + e-iLk] T1i _k3 -CO = 2aUH {T (x) - 2T

(X -

L/2) + (X L) (22) where X -X2 1 sgn x {- X2 - e + + 2X2) erf X} I 1/7

(22')

with

x =

1X1 = /Kx2/4H , as before. Figure 2 shows E(x)/2aUH , as well

as E0(x)/2aUH , for the case L/H = 9 , F = U = 0.2 .

According to slender-body theory, the difference between E(x) and E0(x)

is of higher order than E0(x) . However, slender-body theory is not valid

near the body ends, and so the approach presented above is not necessarily

inconsistent in the sense of asymptotic analysis. In the computation of wave

resistance, the difference is absolutely critical, for, as we have noted, the

wave resistance corresponding to E0(x) is infinite, whereas the wave

resist-ance of the distribution E(x) is identical to the value given by Michell's

integral in the thin-ship idealization, and the latter is not grossly wrong.

ax , 0 < x < L/2 b(x) = a (L -x) , L/2 < x < L , ) -H < z < 0 , we have 2aUH , 0 < x < L/2 Z₀ tx) = - 2aUH , L/2 < x < L 0 otherwise = 2aUH {H(x) - 2H(x- L/2) + H(x-L)}

where H(x) is the Heaviside step function (not to be confused with H(x)

Cr

E(k)

=

0

dx e-ikx[2UHb1(x)] . (21")

Cr

For a strut with the shape of a double wedge of half-angle a , that is,

.

-F,

= 0.2

L/H

9

E(x)

2aUH

ORDINARY SLENDER-BODY THEORY

<MODIFIED THEORY

FIGURE 2.

SLENDER BODY SOURCE DISTRIBUTIONS FOR A SYMMETRICAL DOUBLE-BONDED WEDGE

ADVANTAGES OF USING SLENDER-SHIP THEORY

if appropriate assumptions are made with respect to Froude number, there are two aspects of slender-ship theory that make it potentially superior to

thin-ship theory, namely,

the interference between the bow wave and the bow form is included; the interference between the transverse waves and the entire body is included.

The second is of little or no interest in the strut problem. However, it

should be noted and recognized because it means that one of the primary

short-comings of thin-ship theory is thereby avoided, the so-called "sheltering

effect." This problem has been treated by Reed (1975).

On the other hand, the first may be of great importance in strut problems.

What it means is this: At each cross-section near the bow, waves are created

by the widening local beam. These waves appear locally as the diverging waves

of the usual Kelvin pattern. In thin-ship theory, these waves are all simply

added up. However, in fact, each such component suffers interference from the

other sections of the bow. Thus, the waves which we consider to be generated

at x = x1 have velocity components which violate the body boundary condition

for all x x/ . With the incident stream flowing in the positive - x

direc-tion, this phenomenon is not critical for x < x1 , but it may be very important

for x > _x1 These interference effects are formally of higher order in

thin-ship theory, but they need not be in slender-thin-ship theory depending on the

assumptions made about rates of change of field variables.

A procedure for handling this phenomenon was proposed by Ogilvie (1972), although the only problem worked out in detail by him was based on the extra

assumption that this phenomenon could indeed be overlooked. Such an assumption

was not logically necessary in his analysis; it was only convenient in solving

a specific problem. This phenomenon was included in an analysis by Tuck (1973)

for the problem of a nearly flat body, and it led to significant consequences. The flat-body problem was also partially solved by Maruo (1967). A general

pro 13 pro

14

-cedure for handling these problems has been developed in detail by Daoud (not

yet published). All of these works were effectively anticipated by Cummins

(1956), although he did not solve any specific problems.

The basis of these analyses is all the same, although the rationalization

has taken many forms. Succinctly stated, the problem can be formulated as

follows: On the free surface, the boundary condition is taken as

(1)xx K(1)z =

°

z = 0 , (23)

where, as before,

K =

g/U2 . On the body, there is a kinematic condition,

4

= - Un

1

where n is (strictly) a unit vector normal to the body, and nl is its x

component; for computing the left-hand side of (24), n may be considered as

a unit vector lying in a transverse plane, oriented normally to the contour of

the cross-section in that plane. The Laplace equation is assumed to be

satis-fied approximately in the two dimensions of the transverse planes:

(1,1,17 q,zz = 0

(25)

Finally, it is assumed that there is no disturbance forward of the bow, that is,

E 0 for x < 0 . A formal derivation of this problem was provided by Ogilvie

(1972), and it was embedded into a hierarchy of problems by Ogilvie (1974). The

following discussion largely follows the latter.

A fundamental solution of the problem set by (23) and (25) is:

CO 6 (x) r K r' (x;y,z) - log - H 2 (x) I ek ( z+ c ) cos ky sin (x) x) (26) 7r 7 0

where 6(x) is the Dirac function, r = [(y_T-)2+(,_)2]1/2 ro

[(y_n)2+(z+)2]1/2 , and H(x) is again the Heaviside function. The potential

given by (26) represents the flow due to an "impulsive source" at

(C)

< 0 , if we think of x as a time-like variable. In fact, if we replace

x/U

by t (time), the problem set by (23) - (25) corresponds precisely to a 2-D

problem in which the body shape changes with time (instead of with x ).

A solution of the complete problem given by (23) - (25) can now be given (24)

-Tr

0 0

C(E)

15

-by superposing fundamental solutions, as in (26), over the contour of the body at all cross-section, as follows:

1/2 , 1

(Y-11)2 + (z-C)

1

1

4)(x;y,z)

= -:r7 di a (x;r1,C) log (17-T1)2

+ (z+)2)

C(x) (27) kz x K I dk e

f

cl sin

/a-(x-C)

f

di a( ;71(),C(C))eIg(E)cos k(y-n( ))

where C(x) denotes the contour of the body cross-section at a particular x

and u(x;y,z) is a source distribution over C(x) . A solution is not in hand

of course until a is known, and it must be found as the solution of an integral

equation derived from (24). The integral equation can be expressed in a

rela-tively simply way if we define C(x) as the image above z = 0 of the contour

C(x) and furthermore require that u(x;y,z) = -u(x;y,-z) . Then substitution

of (27) into (24) yields:

1 , 1

_ akxiy,z)

f

di u(x;n,c) n2(y-) + n3(z-C)

2 27 C (Y.'1-1)2 C(x)+C(x)

(Z-02

= _{- Un1} + f(x;y,z)

where

(nn2'n3) = n

, the outward unit vector on the body, and

kz

x K dk ke _sin f(x;y,z) -7 147.1.

)4. 17(x-)

0 0 .1

di ) ekC{-n2sin k(y-n) +_{n3 cos} k (y-ri)1 .

C(C)

The integral in (28) is given a principal-value sense. In (28), the first term

on the right-hand side obviously comes from the right-hand side of (24). The

second term, f(x;y,z) , represents the normal velocity component on the

contour C(x) due to the upstream disturbances. Equation (28) has a simple

interpretation in which the basic character (and difficulty) of the free-surface

problem is removed: We defined a as an odd function in z , for convenience

in writing (28). But this means that a represents a source distribution that

could be used to satisfy a condition cl) = 0 on the plane z = 0 . The compli-(28)

(28')

-,

16

-cations of the free surface have been displaced entirely to the right-hand

side, and (28) is a relatively simple Fredholm equation over C(x) and

E(x)

.

The right-hand side of (28) does contain a inside the triple integral,

and so it may seem that this term should be included with the initially unknown

terms on the left-hand side of the equation. If this were true, there would be

little chance of solving (28). Fortunately, f(x;y,z) depends on a( ;n,,C)

only for E < x . This occurs because of the factor sini47(x-) that appears

in (28'). If we solve (28) step by step, starting at x = 0 , the quantity in

(28') will be completely known at any x . Thus f(x;y,z) properly belongs on

the right-hand side, adding to the nonhomogeneous part of the problem. Of

course, (28) is also an integral equation over the x domain, and, from that point of view, it is a Volterra equation.

This formulation of the problem is actually well-rooted in work early in

the century by Kelvin, Lamb, Havelock, and others. They frequently treated

time-dependent flows in terms of impulsive pressure distributions, which, in

the presence of a free surface, create ongoing fluid motions that can be

deter-mined by methods which are generalizations of the Cauchy-Poisson solution. The

corresponding treatment of steady-motion 3-D ship problems depends essentially

on the transformation of such problems to sequences of 2-D problems, through

application of the slenderness concept. But, after that has been done, the

solution is formally equivalent to the solution of the corresponding time-depen-dent problem, as described above.

Equation (28) permits a direct comparison with the usual thin-ship theory.

First of all, there is no integral equation at all in thin-ship theory, since

the source distribution lies on a plane; therefore the integral term on the

left-hand side of (28) vanishes. Secondly, the right-hand side has two terms.

The first is exactly the same as in thin-ship theory. The second, as already

pointed out, represents the normal velocity component at any section due to the

upstream sources; this term is missing in thin-ship theory too, since the body

boundary condition is satisfied on the centerplane. In determining a so that

both terms on the right-hand side of (28) are accounted for, we have incorpor-ated the diffraction of waves by the ship itself.

17

-A procedure such as the above would not be valid in problems of low-speed

aerodynamics. We have taken advantage of the wavelike nature of the free-surface

problem to treat it almost as if the governing equation were hyperbolic. There is no such basis for overlooking the elliptic nature of the corresponding aero-dynamic problems.

The above procedure is more nearly valid in the free-surface problem as the

disturbance is limited to a region near the free surface. This statement depends

on a result proven by Ursell (1960). The above solution in two dimensions is

equivalent to the true 3-D solution to the extent that we can simplify the latter

to just a system of diverging waves. Ursell showed that near the track of a

translating pressure point the diverging-wave part of the solution (as obtained

by applying the method of stationary phase) dominates the solution. Furthermore,

this dominant part can be evaluated properly by the method of stationary phase

even at points close to the pressure point. Obviously, this is true only because

the solution of the pressure-point problem is so highly singular along the track. The solution corresponding to a submerged singularity is well-behaved on the free surface along the track, but Ursell's conclusion is probably valid to some extent if the submergence is very small.

The equivalence between the 2-D solution and the diverging waves of the

3-D solution is discussed in Appendix A. One point is especially remarkable

and should be noted here: The fluid motion close behind a point singularity

can be treated approximately as a slender-body flow, even though such a

singu-larity appears to violate all of the basic ideas of slender-body theory. Of

course, no such conclusion is possible in infinite-fluid (aerodynamics) prob-lems, and so one may speculate that slender-body theory ought to be more accurate in free-surface problems than in aerodynamics.

METHOD FOR SOLVING STRUT PROBLEMS

In the last section, we showed how a slender-body free-surface problem

can be solved approximately as a sequence of 2-D problems. The procedure has

the advantage that it includes effects of interference between the body and

its own waves. Also, there is no great handicap in treating unsymmetrical

bodies, including yawed bodies (such as struts), which lead to formidable troubles in thin-ship theory.

However, the method has two distinct shortcomings in applications to

struts and a third shortcoming in applications to bodies of extended length

in the longitudinal direction:

It is a slender-body theory, and as such it can hardly be used on

strut-like bodies.

It is based on an assumption of no disturbance ahead of the body.

This is apparently not serious for a truly slender body of shallow submergence,

but it may be expected to be important in strut problems.

Transverse waves are ignored.

We show here how to overcome the first and second objections. The third

will only be discussed briefly in the following section.

Let the potential be written as the sum of two parts:

(1)(x,y,z)

=

epo(x,y,z)

+ ybi(x;y,z) , (29)

where 1)0 satisfies the conditions:

(150 = 0 on z = 0 ; (30)

2(11)r°1 = - Unl , on body; (3L)

ci) (1) 0 in fluid region.

OxxOyy

ozz

The sum of the two terms in (29) must satisfy (23), and so cl), satisfies:

18

-(32) +

-+K

01z lxx =

00z

gC + Ucl)lx = 0 UC - 00z

-

Olz

-

19 -on z = 0 . (33)

Thus, ₀₁ is the velocity potential for a fluid motion which could be caused by the presence of an applied pressure distribution on the free surface. If we let the equivalent pressure distribution be represented by

po(x,y)

, its form is given by

Pox(x,Y) = pUK00x(x,y,0) . (34)

This interpretation is useful in solving for 01 , but it must be used

care-fully. A precise interpretation of (33) shows that there is an imposed vertical

velocity distribution on the plane z = 0 , not an imposed pressure field. The

two separate boundary conditions which combine to give (33) are as follows:

on z = 0

where (x,y) is the free-surface elevation; from the first of these, we have:

(x,y) =

-01x(x,y,0) (35)

It is all right to solve for ₀₁ as if there were an applied pressure field,

given by (34), but we must use (35) to find

0x,y)

.

We also

require1

to satisfy the kinematic body boundary condition:

9n

0 on body. (36)

Finally, we revert to the discussion of the last section to justify an

assumption

that1

is approximately the solution of a 2-D problem:

01yy Cblzz 0 in fluid region. (37)

It should be noted that our conditions are near to ideal for making this

assumption. In particular, the disturbance that leads to the definition of

occurs on the plane z = 0 , and the equivalence between 3-D and 2-D

problems is most nearly valid under such circumstances.

In principle,

pox(x,y)

is defined over the entire plane z = 0 outside

of the body. However, its magnitude drops off very rapidly with distance from

the body. From far away, the fluid motion generated by any translating body

in an infinite fluid appears as if it could have been generated by a

longitudi-nally oriented dipole. When that body moves below a surface on which = 0

the motion appears far away as if it had been generated by a quadrupole. Thus

we expect 4)

0

to have the following form far away from the body: Cxz

= (x2 +y2 +,2)5/2

where C is a constant. On z = 0 , we have

Cx

(POz (x2 +

y2)

5/2 r

and this is proportional to

pox

, by (34). Thus, the disturbance which

generates ₄₎₁ drops off ahead of the body as x The procedure proposed

for determining ₄₎₁ is not valid at great distances behind the disturbance,

and we have a disturbance, pox , which extends upstream to infinity. In

prin-ciple, the procedure violates the conditions of its own validity. However, this

is not likely to cause difficulty, since the upstream disturbance is very small

except possibly just ahead of the body. Furthermore, whenever a smoothly

dis-tributed impulsive pressure is applied to the free surface in a 2-D problem,

the wave motion quickly disperses from that region. Only the possible

occur-rence of transverse waves, not included in this analysis, will vitiate the

solution. This possibility can be safely dismissed too, either on the basis of

actual observation or on the basis of the usual Kelvin wave analysis.

Ahead of the body, the 4), problem involves only conditions (33) and(37). Formally, the solution can be written:

(i)1(xiY,z) = - e dn cos k(y-n)

ITPU VI

pox(,n) siniTt(x-)

, (38)

ck

1 dk kz

I

0 _{x < 0} .

The surface elevation is given by:

(x,Y)

=

-

20 -dk

I

dn cos k(y-n) 1

I

r1)

I

dE

Pox(E,T1)

cos(x-)

(39) 0

x< 0.

Aft of the bow or leading edge of the body, the solution for cl), comprises

two terms, one like (38), a second like (27). In finding the function o to

be used in the latter, it is probably most convenient to solve first for ₄₎₁

-as in (38), the body boundary condition being ignored, then to compute

41/3n

on the body from that solution for substitution into the right-hand side of

(28) in place of Uni after which a second component of qb, may be found by

solving (28) and substituting the result into (27).

The solution as constructed above has some important virtues. First of

all, for a strut-like body, the first approximation satisfies the Laplace

equa-tion in three dimensions. Thus, the solution well below the free surface is an

accurate solution of the actual problem. Also, it can be obtained by existing

numerical methods. Secondly, this first approximation is invalid only near the

free surface, and a slender-body approach is valid there for finding the

neces-sary correction. The fact that the body itself is not slender is irrelevant.

Thirdly, the fact that the free surface elevation or its slope may be discon-tinuous transversely at the trailing edge of the body is not a source of

difficulty at all. The 2-D problems that have to be solved are similar to the

classical Cauchy-Poisson problem, which starts with the imposition of a delta-function elevation or impulse, and the actual surface discontinuities are much

less severe than these. In fact, no special procedure need be introduced for

handling these effects, since distributions of fundamental solutions like that in (26) are quite capable of representing these discontinuous "initial conditions"

and their effects automatically.

-REMAINING DIFFICULTIES

The procedure described above makes no provision for representing

trans-verse waves or their effects. This is probably of little or no importance in

strut problems. However, our procedure could be useful in solving problems,

say, of yawed ships if this deficiency were removed.

This difficulty can probably be worked out with use of the method of

matched asymptotic expansions. A far-field solution would be expressed in

terms of a line of singularities on the longitudinal axis (possibly augmented

by a sheet of singularities'on the centerplane). The strength of the

singu-larities would have to be obtained by matching to the near-field solutions.

This would be straightforward except near the bow, where we have not been able

to devise practical matching procedures. If we obtain a near-field 2-D solution,

as by Ogilvie (1972), we can match it with a far-field solution like that

gener-ated by the source distribution given in (22). The matching is similar to that

described in Appendix A. There is no "overlap region" in the usual sense. Rather,

the outgoing 2-D waves created by a distribution of sources on the centerplane can

be matched to the diverging wave system of the corresponding sources in three

dimensions. If the 2-D near-field solution is not expressed in terms of soarces

on the oenterplane, then the singularity distribution in the far field is not

readily determined. Undoubtedly it will be possible to develop an integral

equa-tion, the solution of which will give the far-field singularity distribution. But

we have not done this, and, until it is done, the transverse waves cannot be

con-sidered along the body in the near field.

The numerical analysis necessary in working out the new procedure will not

be easy, although Daoud has done many of the essential steps and will publish

his material soon. One difficulty may possibly be serious. The solution of the

clpo problem set in (30)-(32) is weakly singular along the line of intersection

of the body and the plane z = 0 . We can see this easily by magnifying a small

neighborhood of this line of intersection, as in Figure 3. For z < 0 , the

normal velocity component is approximately a constant in this small region. For

z > 0 , the normal velocity component has the same magnitude but opposite sign.

If this were a 2-D problem, with

= y+iz ,

the complex velocity potential

23

-in the cross-section plane would be approximately

f = C iC [log - 1]

IT

with C equal to the value of co/ri on z < 0 . The 2-D velocity components

in general are given by:

cpy - ici)z = f'

(c)

= ((log )/y2 +z2 + i tan-1 (z/y)) .

IT

Thus cP has the prescribed values on y = 0 , and

2C

= -

-7-T- log/y2 + z2 .

This very weak singularity appears on the right-hand side of (33), and so the

"effective pressure distribution," pc) , has a singular behavior. In principle,

this should not give trouble, since it is so weakly singular. _{But it may be}

very awkward numerically.

FIGURE 3. SINGULARITY IN (1)0 AT

INTERSECTION OF BODY AND PLANE Z=0

24

-We may note that this singularity could have been avoided had we defined

cp differently. In place of (30), suppose that we had required that

4

/3z

D 0

= 0 on z = 0 . Then (33) would have become:ci) _lz = and the

lxx Oxx '

right-hand side of this condition is not singular in the same way. We did not

choose to proceed this way for the following reasons: For a body of shallow

submergence, there is experimental evidence that the assumption of = 0 on

the free surface ahead of the body is a very good assumption. See, for example,

Ogilvie (1972). For such shallow bodies, our procedure should not require

large corrections to the lowest approximation. This requirement is satisfied

only when we start as indicated in the cpo problem of (30) - (32).

In problems of yawed struts, the most important phenomena may well be

cavitation and ventilation, about which we have said nothing at all. It is

just possible that the first of these, cavitation, can be analyzed by the

pro-cedures discussed in this report. In particular, the appearance of a cavity on

the strut is probably not very sensitive to the actual free-surface disturbance;

if the free surface is replaced by a (I) =osurface, a prediction of the

occur-rence and extent of a cavity would give a good first approximation for practical

purposes. Photographs taken during experiments with struts at the Naval Ship

Research & Development Center strongly suggest that this will be the case.

The prediction of ventilation is another matter. Its inception is probably

the result of flow instability around the leading edge, which certainly cannot

be predicted reliably by any methods such as those discussed here. It is con-ceivable that the flow in a ventilated condition could be predicted as well as

the flow around a cavity, if one is willing to assume a priori that ventilation

has indeed occurred. However, the occurrence of ventilation is so overwhelming

in its effects that the prediction of inception is the most important aspect of

its study. Cavitation, on the other hand, seems to build up gradually, starting

with very small cavities at locations of concentrated low pressure and gradually

enveloping a large part of the strut. From the point of view of predicting the

overall flow around a strut, the details of the cavity flow are not terribly significant unless possibly they provide the trigger for inducing ventilation.

In any case, the theory for predicting cavity flow even in the absence of a free surface needs more development before the free surface is considered more.

REFERENCES

Abramowitz, M., & Stegun, I. A., Handbook of Mathematical Functions, Mathe-matics Series, 55, National Bureau of Standards, 1960, Washington.

Cummins, W. E., The Wave Resistance of a Floating Slender Body, Ph. D. thesis, The American University (1956).

Daoud, N., Force and Moments on Asymmetric and Yawed Bodies on a Free Surface, Report NA 73-2, College of Engineering, University of California,

Berkeley (1973).

Maruo, H., "High- and Low-Aspect-Ratio Approximations of Planing Surfaces," Schiffstechnik, 14 (1967) 57-64.

Ogilvie, T. F., "The Wave Generated by a Fine Ship Bow," Proc. Ninth' Symposium on Naval Hydrodynamics, 1972, ACR-203, 1483-1525, Office of Naval Research, Washington, D. C.

Ogilvie, T. F., "Workshop on Slender-Body Theory, Part I: Free Surface Effects," Report 163, Dept. of Naval Architecture & Marine Engineering, The University of Michigan (1974).

Reed, A. M., Wave Making: A Low-Speed Slender-Body Theory, Ph. D. thesis, The University of Michigan (1975).

Tuck, E. O., The Steady Motion of Slender Ships, Ph. D. thesis, Cambridge

Uni-versity (1963). [See also: "A Systematic Expansion Procedure for Slender

Ships," J. Ship Research, 8:1 (1964) 15-23)

Tuck, E. O., "Low-Aspect-Ratio Flat-Ship Theory," Report 144, Dept. of Naval Architecture & Marine Engineering, The University of Michigan (1973).

Ursell, F., "On Kelvin's Ship-Wave Pattern," J. Fluid Mechanics, 8 (1960) 418-431.

-APPENDIX A: EQUIVALENCE BETWEEN TRANSLATING 3-D SOURCE AND IMPULSIVE 2-D SOURCE

The potential for the fluid motion caused by a translating point source of unit strength in three dimensions was given by (9), and simplified forms

con-taining the essential wave motion were given in (10) and (11). Let us start with

the form given in (11). Under the usual conditions for applying the method of

stationary phase, we find that the integrand has two points of stationary phase

if x > 0 and

fl <

2/-2-x , namely, for k given by

k =

X2 X2

-8y2/x2 .

8y2 - 8y2

The upper sign gives the usual diverging waves, and the lower sign the

trans-verse waves. The dividing value is k =

KI/72-

(for which the inner square root

in (Al) is zero). We note the following in general:

K < k < transverse waves;

< k < diverging waves.

If y2/,2 << 1

, the expression in (Al) can be approximated:

Kx/21y1 (diverging waves)

k =

K(1 +y2/2x2) (transverse waves) .

Then, from (11), by applying the method of stationary phase, we obtain: r-- KX2

K

-

2-1-(1)s(X,y,Z)

=

-/

e 4y21 isin(-Kx2

-L)

7ry 4y 4

/I7

-K1Z.-4-C1

-

sin (Kx + 7/4)

Trx

The first term represents the system of diverging waves. If the source is

actually located on the free surface ( C = 0 ), the approximation for this term

is valid near the entire track of the singularity, even close to the singularity

itself. The same is not true of the second term, representing the system of transverse waves; the approximation for this term is only valid for large values of KX . However, the transverse waves are significant only at a considerable

26

-(Al)

(A2)

--

27

-distance behind the source. (These statements represent a loose interpretation

of results proven by Ursell (1960) for the case of a moving pressure point.)

In the region of interest to us, i.e. y2/x2 << 1 but KX small, the

most important part of the wave motion comes from the part of the integral in

(11) for which k is very large, in fact, for

k/K >>

1. We use this fact as

follows: In (10), which is equivalent to (11) for x > 0 , expand the integrand

for large k by taking

K2

/k2 -K2 = k (1 + )

We then have

= 1

1- 7772

cos (kyl/k2/K2 -1) = cos k2Y

and so the entire integrand becomes k2y

e K sin kx cos

Now let k = , so that this integrand becomes:

k'l _{x cos k'y} and dk = 1-= d k ' 2 V,71TT (10) now becomes: CO cips(x,y,z) _ ek'(V-C) 7 Kki

COS k'y

sin1/57x

.

Since the value of the integral depends only on the higher range of k , it does

not change much if we replace the lower limit by zero. Then this expression

becomes identical to that given by (26) for x > 0 , and the latter is the

potential for the fluid motion in two dimensions caused by an impulsive unit

source at x = 0 .

If the reader is not convinced by the preceding loose demonstration, he

can estimate the integral in (26) by the method of stationary phase, and he will

z+0

28

-obtain the first term on the right-hand side of (A2).

The above results mean that the flow close behind a translating source can be analyzed approximately by treating it as a 2-D flow in the cross-planes. There are some restrictions on this conclusion however:

The equivalence breaks down well before one gets back to the "first" transverse wave.

The equivalence is most accurate for a source at the free surface; it rapidly becomes invalid for any significant submergence of the source.

UNIVERSITY OF MICHIGAN REPORT NO.

175

October 1975

DISTRIBUTION LIST FOR REPORTS PREPARED UNDER THE GENERAL HYDROMECHANICS RESEARCH PROGRAM Commander

40

David W.

Taylor Naval Ship

Research and Development Center

Bethesda, Ed

20084 Attn: Code

1505 (1)

Code 5614 (39)

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Annapolis Laboratory Naval Ship Research and

Development

Center

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21402

Attn: Code 5642(Library)

Commander

Naval Sea Systems Command Washington, D. C. 20360 Attn: SEA 09G32 (3 cys) SEA 03512 (Peirce)

SEA 037 ADF

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12*

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