Recent progress in theoretical studies on the behaviour of ships in a seaway

IlL.

Publishers:

SELVIGS FORLAG

Editor:

PER SELVIG

SHIPBI!IILDIN

NO.4

I

1954

VOL. III

Members of the International

_Edit

Denmark : J. M. Barfoed, B. Sc. director of A/S

Burmeister & Wain's Maskin-og Skips-byggeri, Copenhagen.

C. W. Prohaska, Dr. Techn., Professor, Copenhagen.

Finland: _{Jan-Erik Jansson, Techn. Lic., M.Sc.,}

Head Dept. Mechanical _Engineering,

Technical College, Helsingfors.

France: _{Maurice Terrin, Ing., director of Soc.}

des Ateliers Terrin, Marseilles.

Germany: J. Köhnenkamp, Dipl.Ing., director of

H. C. Stülcken Sohn, Hamburg. A. Weisser, Dipl.Ing., director of A. G. «Weser», Bremen.

Great

Britain : _{AM. Robb, D. Sc., Professor, Glasgow} University.

Italy:

CONTENTS:

Recent Progress in theoretical Studies on the Behaviour of Ships in a Seaway . ₇₄

On Slamming ..

80

Turbulent Friction on a flat Plate

86

On Japanese Progress in Calculation of _{Wave making Resistance} . . .

93

A complementary Method for evaluating Ship Wave Resistance

loo

Theory of Propellers .. .

104

CONFERENCE ON SHIP HYDRODYNAMICS

Shipbuilding of today is a very extensive idea

with a definite scientific shape. _{And a just as}

important as intricate territory within

ship-building is what is comprised under the term

ship hydrodynamics, namely propelling

resi-stance, model experiments, propeller _theory,

speed reduction in heavy seas, etc.

From what can be stated from _exhumations

made, our anscestors in early times had a partly excellent realization in giving their ships lines which made them easy to row and well suited

for shorter trips _{across the ocean.}

To which extent theoretical _{considerations} might have been behind the _{successful result,}

should not be stated, and we may with a certain right say that the classical hydrodynamic is of a relatively new date. Intensive research where higher mathematics and systematical series of experiments have gone side by side have,

spe-cially during the later decades, brought us a

long step further to understand

_{the factors}

which influence the seagoing qualities of a ship.

The main part of this research _{takes place}

with or in co-operation with the different ship-odeltanks all th'e world around. An important

ink in this work is the conferences _{which have}

seen held with 3 years interval and where the

g- n - Head Office: D N DHUSGT. 8, i'OSTBOX 162 OSLO, 11tjY Telep}

G

425509 -412057Telegrams:

SHIPBUIT DING OSLO

leading experts from all over the world within

ship hydrodynamics _{meet to discuss the many} problems which still have_{not been straightened}

out.

During these days the 7th _{International}

Con-ference on Ship Hydrodynamics _(formerly

In-ternational Conference of Ship Tank

Superin-tendents) is being held at Oslo, Gothenburg and

Copenhagen with lectures and _adjoining

dis-cussions.

When we have chosen to dedicate this number

in its entirety to _{an abbreviated repetition of}

the lectures which are held in Oslo it is just to

stress the significance we mean these conferen-ces do have.

Even if only a limied number of our readers

are supposed to be able to read these _articles

with definite advantage, _{one will get a good}

im-pression of what ship hydrodynamics_comprises

as far as difficulties are concerned and which

enormous amount of research work there _is

be-hind the practical results produced. By this we

mean to have contributed to

_{give the right}

understanding of which place this research has

within what we ordinarily _{convey with the idea}

of shipbuilding.

orial Committee:

Holland Pieter Goedkoop, director of Neder-landsche Dok en Scheepsbouw Mij,,

Amsterdam.

O. van den Toorn, Dipl.Ing., director of N. V. Koninklijke Machinefabriek

Ge-brs. Stork & Co., Hengelo.

Ing. Fogagnolo, director general of FIAT Grandi Motori, Torino.

Gino Soldà, Dr.Ing., director general of Registro Italiano Navale, Genoa. Norway: _{Reidar Kaarbo, B. Sc., managing}

di-rector of Bergens Mekaniske

Verkste-der A/S, Bergen.

Georg Vedeler, Dr. Techn., managing director of Det norske Ventas, Oslo. Sweden _{Helge Hagelin, director of}

Uddevalla-varvet, Uddevalla.

K. A. Ringdahl, M. Sc., Associate Pro-fessor, Stockholm.

RECENT PROGRESS IN

THEORETICAL STUDIES ON THE BEHAV

OF SHIPS IN A SEAWAY

by Georg P. Weinblum

I. General remarks.

In what follows we shall report on the

con-tribution made by rational mechanics, especially

by hydrodynamics, to

the study of seagoing

qualities of ships.

This contribution consists essentially in

find-ing the forces to which a ship is subjected in

a seaway and the resulting motions of the

ves-sel. The knowledge of the latter enables us to

calculate for example the acceleration and the

position of the ship in the surrounding water

including to some extent the degree of wetness.

Our present aim is to establish the dependence of ship motions upon its form and weight

distri-bution in the actual seaway or otherwise ex-pressed to furnish basic data for developing

ships with optimum seagoing qualities.

We are, however, far from this goal. The choice of hull forms from the point of view of

seagoing qualities is at present

still more a

matter of opinion than of actual knowledge.

Nonetheless one can list a large number of

purposes for which the theoretical investigations

on the motions of ships in a seaway are useful

or even indispensable. Without attempting

com-pleteness or even a logical order in our enume-ration such studies can yield

A general information on the most impor-tant and characteristic phaenomena of the

behavior of a ship in a seaway.

Prediction of motions for a given ship in a

given (simplified) seaway.

Contributions to the problem of safety by

establishing limiting values of motions, ac-celerations, forces etc.

Explanation of special effects influencing the behavior; e.g. stability, directional sta-bility, resistance.

Establishment of ideas and basic data for

reducing motions (stabilization, damping

devices).

Guidance as to how to plan and to perform

model experiments and full scali research.

74

II. Fundamental investigations on th

oscillations of a ship in a regular seatc

So far there exist three comprehensive

nal investigations:

IOR

There are essentially two branches of

mecha-nics on which our reasoning is based:

hydro-dynamics and the theory of oscillations of rigid bodies.

It must be pointed out, however, that besides

the hydrostatic and hydrodynamicforces which

change rougly with the period of encounter of

the ship other forces of an impact character do arise. These are important especially with high speed craft.

When dealing with this subject one of the

basic simplifications is the introduction of the ideal fluid concept.

Obviously, there exist problems of high prac-tical significance, for example the damping by

bilge keels, where the viscous effects are

de-cisive.

Broadly speaking, however, it is rather

strik-ing how useful the ideal fluid concept proves

to be.

Another basic simplification is the

substitu-tion of regular wave trains for the actual

sea-way.

e

'ay.

origi-The classical memoir presented by Krylov,

a paper by Haskind and one by F. John.

1) Krylov's paper underlies almost all later

studies on the subject. It has been shown

experimentally and by some observations on

sea that Krylov's approach succeeds in

describ-ing the general character of

oscillatory ship

motions in a regular seaway especially when

the ratio À/L is not small. By introducing the

hydrodynamic effects known as added masses and damping a closer approximation to reality is arrived at. On the other hand, several errors are admitted.

The influence of hydrodynamic pressures on exciting forces and moments has been igno-red, especially the reaction of the ship on the seaway has been neglected.

The equations have been linearized.

First order coupling terms have been

ne-glected.

The damping has been treated in a _summary

way.

2) Haskind formulates the hydrodynamic

problem as follows: The displacement of the

ship from its average position is considered as small; therefore the boundary conditions _are complied with at the mean (undisturbed)

po-sition of the hull.

This agrees methodically

with the assumptions made in deriving the free surface boundary condition,

cp

(1)

when the ship is not advancing. The expression

(1) _{becomes more complicated for a vessel}

moving with constant speed if the potential 4)

is referred to axes rigidly connected with the

body. The potential ' (x, y, z, t) studied by

Has-kind consists

of two parts:

the first one

4) (x,y,z,t) represents the potential of the distur-bed motion due to the oscillations of the ship, including their influence on the regular seaway,

the second part is the well known potential

4) (x,y,z,t) _{of the wave motion. By splitting off}

the time factor eic,)t _{since only steady state}

forced oscillations are considered one obtains with

4) (x, y, z, t) = (x,

y, z) eit etc.

"(x,y,z) çc(x, y, z) + (x,y,z) (2)

The boundary condition for 'p (x, y, z) _{on the}

body S is

òp

= V

-òn (3)

where V _{is the normal velocity of a given point}

at the body. From the boundary values (3) and

O the potential can be derived.

We put p _{+ 'po (4)}

where takes care of the reflexion phenomena caused by the ship in a seaway.

Essentially p is calculated by substituting pulsating sources and sinks for the _oscillatory

motions of the body. Kochine has shown that the distribution of singularities over the surface

of the body can be found from

_{an integral}

equaticn and he has proved that for small _and

large values of the parameter k o2/g a

solu-tion exists.

The linearisation of the problem leads to the result that the familiar concept of hydrodyna-mic inertia and damping forces are applicable; they are components of the total hydrodynamic force, and depend upon added masses mij and

damping coefficients N j respectively. Further,

one obtains the usual expressions for the

re-storing terms and formulas for the

exciting

forces and moments Fe and Me:

Fe = _pgeit

(p0+ )fldS (5)

Me = _pgxe1cùt ff [p0=p]rXnds (6)

which by the terms _{'po consider the disturbing}

effect of the ship on the seaway.

The case of the ship at rest (zero speed _of ad-vance) is thoroughly treated _{as a useful} intro-duction to the general case of the ship moving

with finite speed of advance U. _{Added masses}

and damping factors become functions _{of the}

body shape, of the wave length À (or the para-meter k - oi2/g), the course angle x and of the speed of advance U.

Haskind applies his reasoning _{to a study of}

the heaving and pitching motion.

The vertical force Z and the pitching _moment

M consists of four «components»:

Z=Z0+Z1+Z+z3

(7)

with Z1 M11 due to the uniform_{speed of advance,}

Z1 M1 hydrodynamic terms caused by the

os-cillations of the ship,

Z2 M2 restoring (hydrostatic) generalized

for-ces,

Z3 M3 exciting forces.

To obtain explicit results the Michell _(wedge

like) ship is introduced, otherwise expressed, we

substitute for the

_{oscillating ship pulsating}

sources and sinks distributed over the

longitudi-nal center plane.

Leaving aside the constant forces _{Z0 M0 we}

get the following expressions for the

hydrody-namic terms:

Z1 =m33 zm3 1i N33

_{z N53 ç = O}

M1 = - m35z m55 N5 z - N55 =

For the moving ship m35 m53 and

N35 N5

Simplifying further by assuming _{a ship}

sym-metrical with respect to the

_{midsection one}

European Shipbuilding No. 4 - 1954

76

Fig. i

of a «thin» (wedge like) body. Difficult and tedious computations lead to an interesting

ex-tinction curve. Fig. 1. From it we gather that the period concept can no more be sustained rigorously what is not surprising as even the

damped harmonic oscillation is no period mo-tion.

2) Haskind has generalized the problem by treating the three dimensional case, admitting a constant speed of advance,

and considering the coupled motions of

simultaneous pitching and heaving. 2. Forced oscillations in calm water. Theoretical and experimental investigations of forced motions with one degree of freedom in calm water acquire fundamental importance.

They enable us to determine in the simplest

manner added masses values and damping

coef-ficients.

From Ursell's and Haskind's work it can be followed that by substituting source-sink

sy-stems for a body considerable errors in the

de-termination of the hydrodynamic oscillatory

forces (added masses and damping) can be

com-mitted. Recently our knowledge in this field has been appreciably extended by O. Grim.

Like Ursell, Grim restricts himself to the

two-dimensional case of forced motions of a body

with zero speed of advance.

Grim constructs the velocity potential p from:

the potential valid for the motion of a body

in the unbounded fluid,

a term which in a known manner allows to

comply with the boundary condition on the free surface,

a term which enables us to satisfy approxi-mately the normal velocity condition on the boundary of the oscillating body.

Results obtained by Grim for the circular

cy-linder are in close agreement with TJrsell's

find-ings quoted before and other ones

communi-cated in a recent paper which have been reached

by a totally different approach.

In this recent paper, however, Ursell came to rather striking conclusions with respect to the properties of wave damping for various secti-ons. To my knowledge Ursell's suggestion to

check experimentally some of his findings so far has not been carried out.

The results obtained by Grim for added mas-ses and damping forces in the case of the side

motion and the roll are as fundamental as in the case of heaving. Introducing the necessary

m5

N35 = N53

Introducing

Z3 pga A0Ee oet

M3 = pg I'Ve

)et

with » the frequency of encounter, i,, the

moment of inertia of the waterline, the

effec-tive wave slope E and

4t _{the dimensionless}

heaving and pitching functions, the equations of motion are written as

(m + m33)z + N33z pgAoz + m53 + N53

pgaA0Ee et (12)

(J y+m55)+ N55+ pgIlm53zN53z

= pg1y W e

(13)

Thus even for a ship symmetrical with

re-spect to the midship section there is a

hydro-dynamic inertial and damping coupling.

3. Although full credit has been given to ear-lier a paper by F. John, reference is made toit

once more because of two reasons:

here in a lucid form the general boundary

problem has been stated.

Fór waves in shallow water in the presence

of flat floating bodies explicit results have

been obtained.

Following cases solved by John deserve our special interest

Waves generated by a freely floating

cylin-der,

Waves generated by a forced vertical motion of an obstacle,

Wave disturbed by a rigid obstacle,

Motion of a floating body generated by

in-coming periodic waves.

III. Motions in calm Water.

1. Free oscillations.

To my knowledge beside the paper by F. John only two papers dealing with the hydrodynamics

of free oscillations have been published

1) by L. Sretensky: «On the damped

oscilla-tions of the center of gravity of floating bodies».

changes into the structure of the velocity

poten-tial his method is immedia4ely applicable to _the

solution of these difficult problems. 3. Coupled motions in calm water. Following Vedeler the problem of coupling deserve a special paper.

We shall try to supplement the list of coup-ling effects enumerated by him.

The hydrostatic coupling of heave and pitch

is mentioned in any reasonable book

_{in ship}

theory. Denoting the horizontal _distance

be-tween the center of bouyance and flotation by

e, byv and _{the uncoupled frequencies, by i} the radius of gyration of the load _{water line one} obtains the characteristic equation

h4 + (vzi+vd2) h + v2

(le) = O

(14)

where e

y

Because of the extreme smallness of e in most

cases the resulting frequences v+v are very

close to VZVP.

The resulting motions _{are of the type}

C1cos

(+

_{+ ) + C cos (vt + 2)}

₍₁₅₎

z

For free heaving and pitching

_oscillations

there exists the solution found by Haskind. We quote only the simplest case valid for a Michell

ship symmetrical with _{respect to the midship}

section t

(m+m33)cU+ J K[t1

dT dzdr+ o t fK3jt__7)

dr+pgA0Z = 0

(16)

(I+m5s)+cUz +

0 (17) with e a coefficient.

Here K33 K35 . .. . _{are intricate integrals}

which take care of the time history and lead to

complicated motions of the character _{shown in}

fig. 3. From our present point _{of view we are} interested in the coupling terms

_{- cu}

_and

+ cu z. Notice the opposite signs and the fact

that they do not disappear for the _symmetrical

ship except when the speed of advance u 0.

Grim has pointed out that _{the horizontal}

transverse motion and the motion of roll in general are coupled and therefore should be

treated simultaneously.

Grim, also has shown that coupling can exist

between roll pitch and yaw. The theoretical

proof refers to free oscillations in calm water;

it is based on Lagrange's equation _{of motion.}

From the investigation follows that the yawing

motion in calm water is determined by the pitch

and roll. Further it is shown that _{the roll can}

be influenced by the pitching _motion.

IV. Motions in Seaway.

1. _{Exciting forces.} 1.1. _{Theoretical considerations.}

Havelock and the present writer have

cal-culated exciting forces due to _hydrodynamic

efforts experienced by wholly submerged very

elongated bodies moving uniformlyon a straight

horizontal path in a regular seaway.

Hydrodynamic force _{effects are estimated}

using an approximate method due to W. Toll-mien.

Presumably, the application of this interesting

method is superseded to

_{some extent by the}

more general approach due t'o Cummins. We

omit therefore a discussion of the underlying assumptions and state the result, that the verti-cal force Z can be verti-calculated from the apparent

buoyancy force multiplied by the factor

U

i + X33 + [X.j__X11J ₍₁₈₎

provided the ratio À/L is large. The result

agrees with the findings of a more rigorous

ela-borate investigation on the motion of a spheroid due to Sir Thomas Havelock.

Extensive research work on the motions

of bodies in a _{seaway has been performed at the}

University of California.

Restricting ourselves primarily to the theore-tical side of these investigations we mention a

paper by Fuchs and Mac Camy. Oscillations_of

a floating rectangular block advancing _{with a}

constant speed normally to the wave crests in water of finite depth are studied _assuming:

sinusoidal waves,

Stokes waves.

Calculating the buoyancy and moment from the undisturbed wave pressures rather tedious expressions are obtained in the _{second case for}

the heaving and pitching _motions.

By courtesy cf Dr. John Wehausen I had the opportunity to study two recent papers by

M. Haskind.

We shall restrict ourselves to some superficial

remarks on the investigation _{«Oscillations of}

a floating contour on the surface of a fluid with

consideration of gravit.»

European Shipbuilding No. 4 - 1954

a cylinder with a symmetrical contour L

float-ing in a regular seaway. The

hydrodynamic

effects are to be determined. General formulas

are given for the amplitude of waves due to the oscillating body at plus and minus infinity and for the forces Z, Y (horizontal side force) and

the rolling moment M These results are

ap-plied to the well known class of Lewis contours.

1.2. Experimental approach, model

investiga tiorts.

Haskind did not calculate explicitly the

di-stortion of regular waves caused by the ship and

the influence on the exciting forces resulting

therefrom. Together with Riemann he

supple-mented his theoretical research by an

experi-mental method which yields a hydrodynamic

correction for these forces. For this purpose

harmonic oscillations of a model with one degree

of freedom, say heaving, are excited in calm water by an oscillator and added masses and

damping factors derived. Further, the heaving

motion z of the same model is excited by regu-lar waves. Thus the graph

Ze(t5)

(19)

is obtained. Inserting (19) into the equation of

motion

m' z + N33 z +pgA(,z pg A oEei[otzJ (20)

we find a complex

relation from which by

equating the real and the imaginary parts the

exciting force coefficient E and the phase angle

can be calculated. An example of the phase

lag which following the Froude-Krylow hypo-thesis equals zero is shown in figure 2 and the

corresponding force coefficient E is compared

with the «Krylow» value E in figure 3. These

two diagrams are to my knowledge the only

pertinent data so far published.

1.3. A case of gyrostatic coupling.

To my knowledge gyrostatic coupling was treated first by Suyehiro. He deals with the

behavior of a floating body amongst long

regu-lar waves: dependent upon the period of the later the body has the tendency to set itself

parallel or normal to the wave crests. Suyehiro trats the motions of rolling, pitching and heav-ing simultaneously; the equations are linearized

except for a «gyrostatic» term in the relation for

yawing. The exciting term in the equation of

yaw has been neglected, which may be permis-sible when the speed of advance is zero.

78

2. Ship mechanics.

Directional stability in seaway.

Now there is a tendency of treating the

be-havior of a ship in a seaway under the broader aspect of ship mechanics.

The first step in this direction was made by Davidson. The present writer attacked the

pro-blem of the directional

stability

in regular

waves in a more general way. Because of the

extreme complexity of the task this attempt does not go much beyond a formulation of the fundamentals involved.

The special case of a ship advancing in a

following sea has been treated rather

thorough-ly by Grim. Important results have been found which qualitatively at least agree with observed phenomena at sea. I0

o

-Io

-20

E

Dimensionless heaving function E.

2 - calculated following Krylov

i - experimental curve.

A

-.5

Fig. 2

Phase angle between exciting force and the wave.

Transverse stability in a seaway.

To my knowledge, French writers were the

first to calculate the metacentric height of a

ship in a regular seaway from quâsi static

con-siderations.

L

A

The idea was resumed by Kempf and his col-laborators; recently Grim calculated Reed's dia-gram under similar assumptions. The problem becomes now rather urgent in connection with

attempts to «standardize» the transverse

sta-bility.

Using the fact that the metacentric height

changes periodically in regular waves Grimwas

able to clarify an important phenomenon, which so far evaded explanation. It has been observed

that a model running normally to

a train of

regular waves undergoes heavy rolling when the

natural period of roll is twice that of the period of encounter.

3. _{Motions in a confused sea.}

Several times experiments have been made

with artificially produced irregular train of

waves. As motions of models under such con-ditions never reached as high amplitudes as in regular waves of comparable length close to the

synchronism condition the usefuilness of the

regular seaway concept as severest assumption imaginable was corroborated.

Reference is made to a report by Fuchs and

Mac Camy which represents the continuation of an earlier work by Fuchs. It deals with the

heaving and pitching of ships (or models) in

irregular bow and stern waves and is based on

the Fourier integral method, and good

agree-ment is reached between predicted and recorded data.

As the result of collaboration between

ocea-nography and naval architecture a paper was presented by M. St. Denis and W. Piersson. The

part dealing with waves contains _{a thorough}

survey of methods which lead to a representa-tion of an irregular seaway. The statistical ap-proach proposed yields quite new aspects and can already claim a success: Experiments prove

the authors' thesis: in spite of the identity of the frequency of encounter for all _simultaneous

motions of a vessel the number of zero crossings

and maximum amplitudes

over a fixed time

interval may vary appreciable with the motion.

Conclusion..

Summarizing we state that recent theoretical

work in our field has developed in a satisfactory

way. Powerful methods have been proposed,

important special problems have been

success-fully treated, and there exists a promising ten-dency to enlarge the scope of our discipline by

subsuming the theory of oscillations to general mechanics of the ship.

We expect art immediate stimulating effect of

the theoretical work on model research and

later on full scale investigations. Instead of re-lying on «practical» routine model tests only

which frequently are «run» underinadequately

defined conditions emphasis should _{be laid on}

experiments intended to give answer to clearcut questions.

Obviously, our synopsis presents a lot of weak

spots which are partly due to _{shortcoming of} the present writer and partly

_{to the task}

re-quiring a progress report over a definite time. It is probable that valuable _{work has been}

over-looked, some interesting topics _{have not been}

mentioned. The present writer express the hope

that further contributions _{by members of our}

congress will fill out the gaps left by _him

NOTATIONS

A - area of load water line.

E _{- dimension less heaving force function.}

I _{- moment of inertia of the waterline.}

J _{- mass moment of inertia.} K3 _integrals. L _{- ship length.} M - moment in general. N - damping coefficient.

s

- surface. U _{speed of advance.} e _{speed of wave.} g - gravity acceleration. h - lever. = 2 2 k - wave number rn _{- mass.} rn _{- virtual mass.}

m _{- with subscripts: generalized added masses}

(ad-ded masses, ad(ad-ded moments of inertia, mass

coupling factors).

n _{- normal.}

t - time.

a - waterline area coefficient.

- phase angle. - wave slope.

X - with subscripts _{inertia coefficients.} À - wave length. V - frequency. p - density. - time variable. velocity potentia Û) - frequency. - angle of pitch. - velocity potentia

Introduction

The maximum speed of a surface vessel is

not determined by the power but by the ship's behavior in a seaway. Speed of advance must

be reduced to avoid too violent motions. The

most violent attack of the sea on a vessel is

probably the heavy blow delivered by the waves

on the reentering bow. The ship vibrates for some times after such an impact and plates

under the force foot are damaged.

The present paper attempts to describe the

practical aspects of the hydrodynamics of ship

slamming. Many theoretical and experimental results are omitted in order to make the paper more concise, and the present article is only an

extract from the original one.

Definition of slamming

Slamming is felt by the ship's personnel in

the sudden change of the acceleration. The sud-den deceleration is intuitively associated with

high pressures on the bottom and the captain

therefore tries to avoid slamming.

Another

danger signal is the elastic vibration which is generated by the sudden buildup of pressure

(generally called «blow») and which can be

ob-served for quite some time (30 sec, i min.) de-pending on the violence of the slam.

The integrated effect of the pressure on the

bottom is largest at the instant of

slamming,

since the sudden deceleration requires large

forces which come from the pressure on the

bottom plates, but this does not mean that the

pressures cannot be dangerously high immi-diately before or after the maximum

decele-ration is reached.

Contrary to the acceleration record the

mo-tion record do not show any peculiarity at the

instant of slamming.

The considerations above lead to the

follow-ing definition of slammfollow-ing which is applicabl

from the practical and acceptable

from th

theoretical points of view:

Slamming is the sudden change of the

accele-ration of the ship.

ON SLAMMING

by V. G. Szebehely, Dr. Eng.

with the cooperationof M. A. Todd and S.M.Y. Lum, David Taylor Model Basin.

Mechanism of slamming.

When a wedge is dropped on an originally

smooth water surface, the impulse momentum principle can be applied. Neglecting such for-ces acting on the wedge as the buoyance force

the weight of the wedge, friction drag, etc.

very rough first approximation is attained

(ma. + m) V = mV0 (1)

where m is the mass of the ship and ma is the added mass due to the water. V0 and V is the

velocity of the wedge before and after the

pe-netration of the water surface.

The added mass is not constant and can be approximated by the formula

ir p C2

ma=l

₂ (2

where p is the density of water, 1 the length

of the wedge and c the instantaneous semiwidth

of the waterline. It is noted that formula (2)

applies toi a wedge of small deadrise angle and

that it needs correction for the finite length of

the wedge and for the piled up water since the

surface will not remain undisturbed after

im-pact. Roughly, however, the added mass of

a penetrating wedge will be proportional to the

square of the beam at the water surface.

The semiwidth e is proportional to the depth of immersion z and equation (2) may be written

ma k'z2 (k' = const.).

Substituting this value of m a in equation (1) and dividing by m, we have:

(kz2 + 1)V = V0 (3)

where k k'/m.

The acceleration is found by differentiating

equation (3) with respect to time

d2z

Vo2rpzl

a

_{= dt2}

mß2 (1 + kz2)3

where ¡3 is the slope of the wedge. (z = c for

small deadrise angle).

Maximum impact force is associated with

maximum deceleration

T 2

The above simple derivation points out the

essential facts in hydrodynamic impact calcu-lations. If external forces (F) cannot be neglec-ted the impulse momentum principle becomes

t

(m-l-ma)zmVo = JFdt

Differentiating the above equation with re-spect to time one gets

(m+ma)z+mazF

(4)

We see that the force is not equal to mass times

acceleration in an impact process since the

added mass (ma) is a function of time. The _maz

term, or for rotational motion the I_a term,

will be responsible for the sudden changes in

the acceleration.

The time rate of change of the added _{mass is}

influenced seriously by the geometry of a

pe-netrating body. Fine lines cause no large change

in the added mass. The largest sudden change in added mass occurs if the bottom is flat.

Ef-fects such as the elasticity of the bottom _and

the compressibility of the water now becomes

very important.

The use of the concept of variable added _mass

makes our problem one of «unsteady

hydro-dynamics)>, but neglecting _{unsteady effects,} computations lead to unrealistically small re-sults. In summarising it might be repeated that slamming is an unsteady flow problem and its

solution depends on the recognition of the

im-portance of variable added mass.

Effects of slamming

The most obvious and most frequently de-scribed stress generated by slamming

_{is the}

one due to high pressures on the plates under

the fore foot.

The part most susceptable to damage due to slamming is the area of the bottom from 10%

to 25% of the ship's length from bow; in the

transverse direction the keel to 25%' of the beam

is the most dangerous part. Ships of _slender form suffer damage further aft than ships of full form.

It

should be pointet

_{out, however, that}

vibration produced by slamming might also

damage the superstructure. Severe stresses in

light superstructure

may result

in cracked

plates and loose rivets there. The third type of stress generated by slamming increases the

sag-ging stress amidships produced by normal wave action by some 30%. From damage reports, it

seems to be rather certain that riveted ships

suffer more than welded ones; also that gene-rally it is not one slam that causes the damage but repeated action.

Conditions leading to slamming

There are two basic factors which _influence

the slamming tendencies of the ship: th'e lines

at the bow and the velocity and position of the

bow relative to the waves. The increase of beam

with draft is large at the bottom, and the time

rate of change of the added _{mass and added} moment of inertia is generally largest

imme-diat'ely after the bow enters the water. That is the reason why bow out conditon is generally

associated with slamming. A bulbous bow, when

entering the water will result in large changes

in the added mass whereas fine lines _{will act}

to the contrary.

If the bow enters the water _{very gradually,} no sudden changes in the added mass can be

expected even if the transverse sections _{are full.}

Therefore bow emergence is not _{a sufficient} condition for slamming. The phase _between

wave and bow motion should be such that these two oppose each other when the bow reenters

the water. Furthermore, large bow _{velocity and}

large wave velocity are required. _{A small angle} between the wave surface and the keel in the instant of impact will also add to the _slamming. In short, there are three kinematic _{conditons to} be investigated

Bow emergence

Phase lag between bow motion and wave

motion

Magnitude of the relative velocity

Bow emergence

Figure 1 shows the limiting value of the

draft/wave-height ratio sufficient to prevent

bow emergence, with the assumption that the

sea is regular and defined by the _{wave length}

À and the wav'e height h. _{It is interesting to}

point out that if the ship is in _{hove-to condition,}

(F _{O on figure 1) the bow might still emerge.}

In fact, for relative short _{waves (À/L .79) bow} emergence is more probable in hove-to condition

than if the ship is advancing.

Phase lag between bow motion and wave

motion

Figure 2 presents the _{phase-variation with}

European Shipbuilding No. 4 - 1954

e

82

lb.r

Fig. 1.

Bow does not emerge if the limiting value of draft!

waveheight as obtained from the curves is smaller

than the actual H/h Wavelength parameter: y = ,.L/A.

a

I,

7To. Ir

Fig. 3.

diagram one might conclude that there is a

cor-responding dangerous speed for every wave

length for a given ship, namely when the bow

motion and wave motion oppose each other.

Relative velocity variation with Froude's

number is shown in figure 3, and one may see that also in this respect there are certain speeds

which are critical.

Conclusions regarding the sea conditon and slamming

From the above discussed role of the three

most important factors and the figures presen-ted, we arrive at the following conclusions:

If the wave length is approximately equal

to the ship length (A 3) the speed

correspond-ing to Froude number .1 is dangerous.

In long waves (A = 1) neither the bow

out condition, nor the magnitude of the relative

vertical velocity, nor the phase between bow

and wave motion is critical.

In short waves (A 4) the hove-to

con-dition or slow speed are the most dangerous

and generally a speed increase reduces the

probability of slamming.

Very roughly in the conditions for bow emergence, large relative vertical velocity is

associated with opposing wave and bow velocity. The complete picture is more complicated,

how-ever, since for instance in waves 57 % longer than the ship (A = 2), the critical Froude

num-ber from the point of view of bow emergence

and relative velocity is approximately .3,

therefore either a speed increase or a decrease

would reduce the probability of slamming.

From the point of view of phase between bow

and wave velocities, the F .3 is not as

dange-rous as the F .4 or .5 speed, therefore the

previous recommendation is to be modified to

say that in these waves a

speed reduction is

definitely recommended.

Hove-to condition might easily result in

slamming. The most dangerous waves are

ap-proximately of the same length as the ship.

The probability of bow emergence

in-creases if the draft is reduced, keeping other

conditions the same. If this happens the impact

velocity might be of the same order of

magni-tude as the ship velocity.

Method of slamming computation in waves

Motion predictions of ships seem to

be in

a rather preliminary

state, even if regular

waves are assumed. Slamming depends very

2

r

0 .1 1.57

Í1.1

7/L. 3.1. .2 .5 360 2 _______________ 4.-..79 1.-3 12 o.

'i

8 1

H

.1 o .1. .4 1.-t. 2 ?roude Jube r Fig. 2.

If

-

l80, the downward moving bow meets

an upward mowing wave. Wavelength parameter,

7 = L/A.

E'

p. a. .4 ap

European Shipbuilding No. 4

- 1954

for the real ship form in

_{the hypothetical}

motion.

The differential equation of _{motion is now} solved using the above obtained variable

coef-ficients. The new equations _{are written in the}

following form

(6)

[I+Ia(t)]

_{+N(t)+ia(t)1T=Mqi(t)}

[m + ma(t)] + N (t)7 + nia(t)z

_{= F(t)}

Attention is called to the added mass ma, the

moment of inertia _{a and the damping factors}

N and Nçi now expressed as functions of time.

One may also notice the _{new terms ii az and 'a}

containing the time derivative of the added

mass and of the added moment of inertia. _Since

slamming is due to the sudden change in these quantities the above terms will determine the magnitude of slamming.

Solutions of equation (6) satisfying _the proper initial conditions can be obtained _by numerical or graphical methods. _{These new}

solutions will be different from the originally

assumed simple harmonic motions. _{The bow}

acceleration is _{strongly influenced, whereas}

the velocity curves and especially _the

displace-ment curves might differ _{very little from the}

originally assumptions.

This second approximation _{can now be}

con-sidered an _{improved representation of the}

ment curves might differ _{very little from the}

procedure, one can again make new drawings of the ship in the waves, based on the new

so-lution.

This step corresponds to step (3). The new added mass, damping force, _{etc. variations with}

time are obtained from the drawings and the new differential equation is established.

Corresponding to step (4), the new

diffe-rential equation is solved and _{the third}

approxi-mation is obtained. The above process can be repeated until no significant _{difference between}

successive solutions is obtained, _{but the labour} involved in this method is tremendous. Figure 4

shows measured and _{computed bow}

accele-rations. The computed _{curve does not show the}

high frequency elastic vibration _{picked up by}

the accelerometer.

Pressure on the bottom

From structural point of view the pressure

distribution and its time variations are

impor-tant. Large pressures which last for micro-seconds are of no practical _importance. strongly on phase relations between wave

motion and ship motion and slamming generally

is the result of violent ship motion. If the

pe-riod of encounter is close to the natural heav-ing or pitchheav-ing period, the phase lag _predictions become extremely uncertain, since in this case

damping strongly influences the motion. In fact,

motion with very small damping has _{zero phase}

lag below resonance, and 1800 above resonance.

Around resonance the amplitude is also _very

sensitive to damping.

When coupling is neglected the _governing equations for heaving and pitching motions

might be written as

(m + ma) + Nz + Apgz

= F COS et (5)

(I + a)

+ N

+ J P g =

Sfl CUet

where 4' is the pitch angle, A _{the waterplane}

area and J the moment of inertia of this.

If the ship is assumed to be wailsided there are methods for estimating the coefficients F

and M

_{(unbalanced hydrostatic force and}

unbalanced moment). There are also ways to

estimate the added mass and added moment of

inertia. The paper also gives reference to

me-thods for the estimation of the damping

fac-tors N Z and N.

However, the ship most certainly cannot be

considered a wall sided vessel if part

of it leaves the water, as is generally _{true in}

slam-ming. Coupling between pitch and heave cannot

be neglected either, since phase _{relations are}

strongly influenced by coupling _{effects. In the} following a proposition is made regarding an explanation and computationel method for the slamming phenomen where attention is paid to the circumstances indicated above:

The pitching and heaving motions are

estimated from equation (5) assuming _{that the}

vessel is wall sided, that the _{damping has its}

steady state value and that the added _{mass and}

added moment of inertia of the ship _are

con-stant, etc.

For one cycle, at suitable time _intervals, the wave surface is drawn with the ship in the waves, in the position which can be computed

from the previously found and phase lags.

Now the drawings are used to determine

the bouancy force and moment _{for numerous}

instants. Also the waterplane area, moment of inertia of same, the added mass and the added

moment of inertia can be found by means of these drawings. Thus the time _{dependence of} these quantities is approximately _determined

Eurcpean Shipbuilding No. 4 - 1954

84

Fig. 4.

It is known that the slope of the transverse

section under the fore-foot and the velocity of

the impact have more influence on the

maxi-mum pressure than doesthe acceleration. Since

neither the bow velocity nor

the motion is

influenced very much by slamming, an

approxi-mate pressure distribution calculation might be

based on assumptions involving harmonic

motion. The pressure at the keel is mostly in-fluenced by the relative acceleration. Therefore at the instant of slamming, pressures are to be

computed at the keel as well as at other

loca-tions. It can be expected that at some locations

the pressure reaches very high values before

and after slamming. The magnitude of the area of the high pressures is of importance as well

as the time

duration of the pressure peaks.

Pressure distributions for several instants

be-fore and after slamming should therebe-fore be

computed.

From theoretical unsteadyflow considerations one can arrive at the result that the location of

the maximum pressure is very near the spray root. Figure 5 shows a transverse section and

the location of the maximum pressure point.

Figures 6 and 7 show pressure distribution on the bottom of a slamming Liberty ship.

Slam-ming occurs at t 1.285 second after the wave

iiiI/11'

ON T BOroK Pr. s surs Fig. 5. PRESTRZ DISRIBtÍrIcN pray 1ed up Water Undisturbed water Surfacs Point or flaX1.0 Pressure

crest is at the midship. Figure 7 gives the

pres-sure distribution a short time after slamming. The two figures illustrate the fact that the

mag-nitude and location of the maximum pressure

vary considerably during the impact process.

Experiments

A total of 200 experiments were performed

with a 5'/2 foot model of a Liberty ship. Two different draft conditions were used. Speed

reduction curves obtained in regular waves of

A/h 23 at ballast condition for various tow forces, are shown in figure 8.

From this figure it might be pointed out that

the relative speed loss is smaller at high still water speeds than at low still water speeds, or

that a low-powered vessel encounters more speed loss in a heavy sea, than does a

high-powered one. Another characteristic of the speed

reduction curves is the shift of the maximum

speed loss toward higher wave lengths as the

tow force or corresponding still water speed

is increased.

Experiments showed that the relative speed loss percentage wise is practically independent

of the draft. However, model in ballast

con-dition would respond more violently in pitch

under identical wave condition and thrust. Bow

out condition is more easily established at re-duced draft than at heavier draft corresponding

to normal displacement. Violent motion plus

probability of bow emergence due to reduced draft go hand in hand to produce larger

slam-ming.

In conclusion it might be pointed out that

successful slamming experiments can be

per-formed in regular waves if the d.raft is

suffi-ciently reduced and if higher than design speed

is used. If the purpose is to present a physical

picture of the slamming phenomen, the use of

high speed is justified. COMPARISOf AO COMPUTEO BOW £CCELERATON OF MASLJRD VtflCAL

L

2 Jr 2 I 6 to to

Di.t a3 _{b. fris tisi (f t)} Halfbeam: 28.5 ft.

Fig. 6.

The pertinent factors influencing slamming,

and listed in the theoretical part of this _paper,

were also found experimentally.

There will easily be variations in the

experi-mental results, but this is not necessarily to be

attributed to experimental

_{errors, due to the}

fact that slamming introduces _{non-uniformity}

in the motions. Slamming depends _{very strongly}

Distar. .ioij tias tris D.1 (ft.) Halfbeam: 28.5 ft.

Fig. 7.

on phase relations and it often takes place near resonance. The slightest changes in the

experi-mental conditions might influence the phase

relations and, therefore, slamming.

Repeatabi-lity of slamming experiments can be considered

good if the successive experiments performed

under identical conditions result in less than

10% deviation, considering average values.

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86

TURBULENT FRICTION ON A FLAT

PLATE

by A. A. Townsend, Emmanuel College, Cambridge.

1. Introduction.

Until fairly recently, the search for an

ade-quate theory of turbulent shear flow had to be

conducted without detailed information about

the structure and mechanism of turbulent flow,

and, although the physical insight of L. Prandtl

and G. I. Taylor led to remarkable progress in

the description of the mean properties of the flow, this lack of knowledge of the turbulent motion itself led to incorrect assumptions about

its nature that produced inconsistencies in the theory. The development of a theory of

turbu-lent shear flow based on an exact knowledge of

the turbulent motion may be traced to the

in-troduction of the hot-wire anemometer, which

made possible accurate measurements of the turbulent velocity fluctuations, and to the

for-mulation of the statistical theory of isotropic

turbulence by G. I. Taylor, who showed for the

first time that an exact treatment of a turbulent flow based on the Navier-Stokes equations of motion might be possible. For some time after

this beginning, theoretical and experimental

studies were confined to isotropic turbulence,

but, as the understanding of this simple flow

grew, several workers began to make detailed

measurements in turbulent shear flows and to

seek regularities in their behaviour. As a result of this work, the structure of turbulent motion

has become fairly clear, and a start has been made in the identification of the processes which

cause turbulent flows to maintain their

charac-teristic levels of shear stress and

turbulent

intensity. For free turbulent flows, it is now

possible to make approximate but absolute

esti-mates of the rate of spread which are in good

agreement with observation.

The purpose of the following account is to

review briefly the general characteristics of free

turbulent flows and of channel flows, and to

interpret the flow in a boundary layer in terms of these two types of shear flow.

2. Notation.

Rectangular axes are used, so chosen that O

is the direction of the mean flow close to the

plane, y O, which is a plane of symmetry in

a wake and a solid boundary

in channel or

boundary layer flow. Th'en

u, y, w are the components of the

tur-velocity,

u, y. w are the components of the

tur-bulent velocity, is the mean pressure,

is the kinematic viscosity of the fluid,

is the constant mean velocity in the free stream,

is the stress in the Ox direction

across a plane with normal

pa-rallel to Oy,

ro

is the shear stress at the wall,

y = O,

is the total thickness of a boun-dary layer,

R U1X/v is the Reynolds number

descri-bing the flow at distance x from the leading edge,

is the local resistance parameter,

are constants in the

universal

velocity distribution in the con-stant stress layer ('equation 4.5), is a function describing the ve-locity distribution in a boundary

layer,

are constants,

is a constant defined in equation

(7.1),

is a constant defined in equation

(9.6).

should be noted that the pressures and

stresses used in this paper are «kin'ematic»,that

is, they are the usual mechanical ones divided

by the fluid density.

3. Free Turbulent Shear Flow.

Free turbulence is a term used to describe

turbulent flows which are not restricted in any

direction by rigid boundaries, the

principal

shear flows in this group wakes, jets and free

mixing zones. All these flows are very similar

in structure and in dynamics.

The first and most fundamental characteristic

of all turbulent flows is that they are

statisti-= -

l/2/rj

K,A

P

U' xy