Non-Kelvin dispersive waves around non-slender ships

Non-Kelvin Dispersive Waves around Non-Slender Ships

K. Eggers.

TEVU$HE UVZTT

abomtortum vcor

Sthoep$hydmmetha1ca

kcMet

Mekeiweg 2,2828 CD Detft

ieL Ot - - Fae O18 181833

I

Introduction

The corrnon assw7iption that energy

of

waves on a non-uniform current is propagated with iCd where Ca

is

the group veloc-ity, and that no further interaction takes pTace, is sTown in this paper to be incorrect.

The above statement may be etected in the introduction to a paper of

Longuet-Higgins and Stewart

J

I . - Within the analysis to follow, we shall introduce

a modification of Kelvin's classical dispersion relation,which leads to a sub= sequent definition of group velocity, which in turn restores the correctness

of the above common assumption. - In an earlier paper 121 Of these authors, the introduction reads:

Short gravity waves, when superposed on much longer waves

of

the same type,

have a tendency to becOme both shorter and steeper at the crests and corre=

spondingly longer and lpwer in the troughs.

We shall verify this statement for the case that for "much longer waves" we

take the flow field around the ship in the limit of gravity tending to infinity, i.e. the so called zero Froude number

double-body flow. It seems to be

essential, however, that the wave propagation is investigated in horizontal planes rather than onisobaric surfaces of the double-body flow field (which

anyhow will

not

correspond to stream surfaces).

It appears to be meaningful to subdivide such a plane into areas where a certain partial differential operator regarding the horizontal variables is either "elliptic" or "hyperbolic". We observe a widening of the Kelvin angle

(measured against the local inclination of double-body stream lines.) near bow

and stern, i.e. in the hyperbolic region, whereas the wave crests appear to be bent towards the center plane in the elliptic domain aside of the middle body.

This is well in accord with experimental observations.

If q stands for the dimensionless modulus of the flow vector in a ship-oriented system (such that q'l far away from the ship), we find that the hyperbolic do main is determined through q less than nity. If U, stands for the ship's speed,

p for the fluid density and if p = p U (1-q2)12 stands for the pressure under

neglec of the gravity component, we may introduce a (critical?) reference velo=

city c = . We shall show that for wave coponents either both the phase- and the goup velocity are different from c and from each other, or both coincide with c'. This is in a certain analogy to the case of waves generated by a ship cruising at supercritical speed on water of finite depth, but in our case the direction of waves is bounded from above through a Mach angle"

N =

arc sin (c /Uq), as phase velocities generally will exceed c

In the "elliptic" region q2>l we observe another restriction on the wave propa=

ation angle from above, which may be of direct relevance to the incipience of wave breaking, if we accept the physical principle that a wave pattern can build up only in those areas for which the associated group velocities are directed away from the source of energy.

In discussion to a paper of Yainasaki 3 a boundary condition for the undisturb ed free surface z = 0 was derived from the requirement that the wave resistance,

in the form of a quadratic functional on flow and wave elevation on a control surface surrounding thE ship.up to the water surface, should be invariant from

choice of such surface up to the sixth power of the Froude number. We shall now

investigate the existence of periodic wave solutions to Laplace's equation in the lower half-space which satisfy the above boundary condition

locally,

i.e.

under the assumption of conàtant coefficients for this partial differential

equation. We thus accomplish a refined basis fOr ray tracing methods as investig=

ated byInui and Kajitani

II

and by Keller ₁₅₁ more recently, where the "exit=

ation" i.e. the r.h.s. D(x,y) as introduced byBaba 16,1 is taken as zero. It will be shown that

(1) The maximum possible wave length increases proportional to q2-1/3, and

for q2 < 1/3, namely within small vicinities of the stagnation points, waves cannot eist.

There is a minimum possible wave length, proportional to q2-1,.in the region q2 > 1. This domain Extends sidewise from the ship to infinity.

Along the boundary lines q2 = I we retrieve the classical dispersion

relation of Kelvin and Havelock, but it is only here that the maximum

wave length coincides with that for a parallel flow ofmodulus qU.

Referring the wave propagation angle y and the coordinate angle

a

to the local

tangent direction of double-body flow, it is further shown that

The modulus of y is bounded from above by the requirement that

cos2y

must exceed both -1/q2 and 1/q2 -

2.

There is an upper bound arn to the modulus of those coordinate angles

a

for which not all wave components extinguish each other through inter= ference (i.e. there is a certain y for which the phase is stationary),

and grows along with 1/q2 up to

rrI2

for 1/q2 = 3.

Whereas for q2>'\0.7 for any

a

meeting above restriction we have two stationary.yvalues (representing so-called "transverse" and "divergent"

waves), coinciding in the casE

a = am

_max

will

be refered to as the

Kelvin angle al( in this case), for smaller q2 we find only one station ary y value, which coincides with

ii/2-a

for

a =

In this case, _ctm

is the "Mach angle" aM as defined earlier. The situation is similar then to that of supercritical shipwaves, where the wave spectrum is composed of divergent waves only.

For all wave modes, there is an exponential decay of the amplitude in the

direction of the gradient of q2, i.e. with decreasing basic flow pressure.

This, and the above mentioned increase in wave length with q2, is in accord

with the analytical findings of Longuet-Higgins and Stewart

_121.

2. The boundary condition on the plare z = 0

The basic assumption of slow ship theory is that for low Froude number the actua]

flow components may 1e considered as results of "small" disturbances to an ideal double-body potential flow in the entire fluid domain, and there is some experi= mental support for such claim at least near the ship's forebody. Another assumpt

-224-ion is that the disturbance velocity potential admits 'a uniform expansion with regard tO the ship's speed U.

Rather than seeking for a sophisticated justification, we shall axiomatically

make an al'ternat.ve assumption, namely that and its derivatives vanish as strong as.O(U ) with U approaching zero (if notstronger). We shall confirm then that this assumption does not lead to formal contradictions. We derive a

boundary condition for on the plane z 0 which admits locally periodic solutions,, though an "outgoing wave type character't is not achieved autornatic=

ally. This is in accord with the fact that we may satisfy the free surface

I boundary condition up to any power of U with flow models void of far field

waves, say by distributing Rankine sources.

We shall prove, however, that if the above boundary condition (together with Laplace's equation) is observed, resistance derived from control surfaces is

invariant at least up to terms

0(U6),

and from the known rate of decay of

double body flow it is apparent then that such resistance vanishes as strong as 0(U6) at least with U. In fact, the above boundary condition will be

derived from this requirement.

-Let stand for the solution of the exact boundary value problem of stationary ship wave potential flow in a frame fixed to the ship, and let st'and for the double bodyflow potential with flow components u,v,w. Let = cr+ 3, and

= 0(U3). Then w, w, wy, and hence u and vz vanish at z = 0 for reason of symmetry. The exact free surface condition Dp/Dt = 0 for z = r(x,y), where p

is the pressure and

= (U2'-(u+)2 (v+

may then be written

=ug

+v

g(w+) + ((u+

)/B

+ (v+

)/

+ (w+

x

, y y

2

(w+ )2)/2g is the wave elevation (compare Newman

7W

)/

)((u+

)

z x

(2.1).

Satisfying this equation up to terms 0(U5) only, replacing w.through w within

such approximation and shifting terms not containing to the r.h.s., we obtain

+ u2 + 2uv

_{+v2yy+x(uux+ivy)}

_{+ yx+uy)}

w

+ o(U)

(2.2)

rx ry

rz

with = (U2 - u2-v2)/2g, u and v (as in the sequel) evaluated at z = 0. 'Observing

that

_{Wz_(Ux+Vy)}

=

(x,y,O) +

_rzz

+ o(U5)

and that with error o(U5)

'

at z = Crmay be replaced by'

values for z = 0, we have

(u2-g

)d'

+(v2-g )4 +2uv -2g(r q + ₎ + g4 g(u ) + g(v )

r xx r yy xy rx x ry y z r x r y

(2.3)

Save the terms connected with on the l.h.s. of (2.3), this equation has been derived earlier in the analyses of Baba 6[ and of Newman I. The extinction

of the terths and -2 is achieved there by stipulating a change of order by U for

horizontldifferentiation of

. We shall see that through

such manipulation the homogeneous part of (2.3) then admits (locally) purely

- 225 -' SchilTstechnik Bd. 28 1981 )2)/2 = 0.

periodic solutions, whereas otherwise an exponential. variation of amplitudes must be taken care of. -As it is by no means evident that must have only-wavy character even in the near field, this order change appears as a rather arbitrary confinement withOut providing benefits through a significant simplification o

the analysis.

The vanishing of the rterIns linkecLto the second derivatives of p in (2.3) is

equivalent to satisfying the boundary condition on the curved isobari.c -surface

z

= rather than cn the plane z = 0 within errors 0(U5) (see Newman

171).

Baba ₆₁ meets (2.3)on z = 0 at the expense of invalidating Laplace's equation 'in the lower half space through a vertical coordinate stretching,.

In Keller's ₅₁ approach, is decomposed as = + iii', where i satisfies = Dr(x,y) on z = 0 with Dr (rru) + (rv)y , and ' is postulated 0(U4) save an exponential factor which is noanalytic with U approaching zero. For the

differential equation ij' has to satisfy, it is not evident that any invariance

of resistance is secured up to 0(U6), but through this special approach there is obviously some "Outgoing wave type behaviour" impressed on .

We should observe that for a ship with fore and aft symmetry , the only influence from a change in the ship's speed from U to - U on eq. (2.3) is felt in a change

of sign of Dr(x,y), thus -(x,y,z) would be a solution for this case. But if the

potential shows waves only

behind

the ship, - cannot be the proper solution

for the case of the ship traveling astern, it has to be supplemented by an ad= equate combination of solutions to the homogeneous part of (2.3) i.e. with D(x,y) set equal to zero.

Uniqueness of the boundary value problem posed for may be achieved if we require that for any transverse plãne ahead of the ship associated wave resi=

stance integral as defined in ₇₁ should vanish up to o(U6), but it is not clear

how this can be verified numerically - if we could solve the problem by a Green

function approach, it would certainly do to satisfy a radiation condition for such Green function (which would be a genuine function of 6 variables, coined by the double-body flew). Otherwise, only brute force methods to secure the

proper far field behaviour of qi seem to be available up to now.

-3. Derivation of a dispersion relation

We derived above the boundary condition

(u2-g

) + 2uv- +- - 2g -2g +

- - r xx xy y

rx x

ry

gqa gD(x,y),

(3.1)

to be satisfied on the plane z = 0.

-- Let us consider a vicinity of an arbitrary point P

= {x,-y, O} where the

derivatives

_VxVy

may be considered small in the sense that (3) may be

considered as having constant coefficients (as given for P ), and let us set

D(x,y) = 0 there.

-Introducing the principal wave number k = gIU2, let us consider a potential

A exp{kK(1+ilJ) (z + i(x-x )cosO + i(y-y)sinO)} - Introducing polar coordinates R and 6 through x-xR cos 6

(see fig. 3), we have

-= A exp{kK(1+iT.1) (z + iR cos (B-&) } (3.3)

(3.2)

-yR sin 6

-.K0/(2sq). .0 1.06 rc) Fig..2. :

7Zz'ric dornn

Vari3t1 of q in case of an ellipsoid

Fig. I.: _{Dependence of dimensionless wave1enth on ave}

angle v (uasured against double body flow) and on q2.

S '5 _N

30 - - 0 &o 10 00

(Ni'Lcrs OiC.' Zincs ahc' t-on tines q2)

N 'S S. '5S. N

-'

-S_ 2C c.15 5'. q'=1.00 ,. .. . _C96 C 90 0.9.

Z2d

with a.r-ia rtior.F 1.01, 1 02 - 227. - Sehifistechnik Bd. 28 1981

With the attenuation term i

set equal to zero, (3.2) and (3.3) ar

represent=

ations of a plane wave of length A =

2rr/Kk0 an4 angle 0 of its normal against

the x-axis. Introducing polar coordinates in the velocity (hodograph) plane

as well, defining

through u

Uq cos1, v = Uq sine, and inserting'

(3.2) into

the boundary equ'ation (3.1), dividing by K(1+ip), we obtain

Separating real and imaginary part, introducing y =

O-$as the wave angle

against the double-body flow direction.and the wave, number

k = k0K we find

k/k0 = K

2/((1 + 2cos2y)q2-1)

2iil(k6A)

= -2(

cosO +

sine

)

= - 2---r

rx

'

ry

R

c0

const

The requirement that k must be positive to secure

downward decay of the wave

implies the followi-ng restrictions on wave length and on wave angle:

27r ₍

(i)

0<A<

(q2-1

k

2 0

4. Systems of waves,

Schifistechnik Bd. 28-1981

3q2 - I)

2 <A<

228

-'for 1/3

<q2<1 2ir

(3q2 - I)

'

'for q2>1

' '

(3.7.)

k

"-

2 0

We shall find in the next section that if we exclude waves

with negative group

velocity, the limitation from below in (ii) must be enforced and that a re

striction similar to (iii) must be applied to the'wave angle for

q2>1

(iv)

For t'he domain where q2<1/3, i.e. near the stagnation points,

where

'Tj

exceeds Uq, no stationary waves can exist.

We may conclude from (3.6) that there is an amplitude .decay with decreasing

i.e. with pressuredecreasing in the direction of wave

propagation.

The dependence of Xon y for various values of

q2 is displayed in fig.

I

The location of lines where, q2

I

and

42

= 1/3 is shown in fgs. 2 and '3 for

the case of the Inuid

S 201 and--for a semi-submerged .11ipsoid with axis

ratios 1:O.5:0.25.

Kelvin' angle and stationary-phase relations

With polar coordinates R,a oriented to the local double-body flow

i.e. with c=ô-

as visualis'ed in fig. 3 we have

x.,y,0) = A exp{-k(y) ii(y) R cos(cz-y)}expik(y)Rcos (a-y)}

with

-

-2r0S+4 crysincnI

direction,

(4.1)

(42)

kU2{K(1+i.1)(gC/U2

_{q2cos2(0) ) +(I-2i(}

_{cosO' +}

ryS1h1E ) )} = 0'

3.4)

-

(3.5)

(3.6)'

zuverIssig und wirtschaftlich

Die Betriebsergebnisse unserer TM 410 urid 620 Hächstleistungsdieselmotoren haben ihre Zuverlassigkeit und

Wirtschaftlichkeit béwiesen.

Sie sind besonders in Hinbiuck auf die Brennstoffe der Zukunft konstruiert

Die lnvestitionskosten für em Schiff oder eine Kraftstation werden erheblich gesenkt. Die Betriebskosten eines Schiffes sind niedriger rn Vergleich zu einer veralterten Antriebsanlage

Die gewinnbringenden Teile des Schiffes (Laderaum) werdén vergrässert-durch die kompakten Abmessungen derMaschine.

- Die Wartungskosten sind niedriger wegen der guten Zugänglichkeit des Motors und der grossen Wartungsintervalle.

Separate Ventiltrieb-schmierüng deshaib keine Verschmutzung -des Triebraumôles. Intensiv gekUhlte AuslaBventilgehause. Keine Ventildrehvorrich-tungen erforderlich. Keine

"Ventildurch-brenner' bei Verwen-dung von Brennstoffen

mit hohen Vanadium-gehalt. Doppelbodenkónstruk-tion desZylinderkopfes. Steifer Zylinderdeckel. Gunstige thermische Belastu ng. Geringe Verformungen durch den Verbrer,-nurigsdruck.

TM 410 und TM 620 hervorragende SchwerOleigenschaft durch

TM 410 und TM 620 Dieselmotoren mit einem Leistungsbereich von 3.385-16.180 kW (4.600 - 22.000 PS). F 240 Dieselmotoren 810- 1.470 kW (1.100 - 2.000 PS). DR 210 Dieselmotoren 235675 kW (320-920 PS).

Stork-Werkspoor Diesel

Postfach 4196. 1009 AD Amsterdam, Holland. Telefon (020)5203911. Telex 11513 swd-nl. Postfach 608, 8000 AP Zwolle, Holland. Telefori (05200) 71717. Telex 42116 swd ni.

Büro Hamburg: Ost-West Strasse 67, 2000 Hamburg 11. Telefon (040)371256:57. Telex 02- 16 18 78, swd d.

- 229 - - Schifftechnik Bd. 28 - 1981

Doppelte EinlaBkanäle. Zur Verhinderung uner-wünschter

Verwir-belung wâhrend des Aufladevorganges.

Voraussetzung für gute Verbrennurig von Brennstoffen mit hoher

Conradson Carbonzahl.

Steife Zylinderlauf-buchsen mit hohem wassergekUhlten

StUtzring. Exakt runde

Lauf-buchsen unter allen Lastbedingungen. Separate Laufbuchsen- -schrnierung.

Verhindert

Kaltkor-rosion in der Maschine

bei Verbrennung von

-Kraftstoffen mit hohem Schwefelgéhalt.

Schiffalirts-Verlag ,,Hansa" C. Schroedter GinbH & Co. KG

Aus dem Inhalt:

Entwicklung des Schifibaus zU den heutigen Schifistypen

Fabrgaststhifie - Fahren - Fraditsthiffd für feste Ladungen .- Linienftacbter - Containerschiffe - Ro/Ro-Schiffe - Barge Carrier Trampfrachter Maasengutsthiffe Erz/Ol und OBOShiffe Tanker Produkten und Chemikalientanker -Gastanker - Spezialschi.ffe - Autotransporter - Arbettssthiffe - Fischéreifahrzeuge - Offshoretechnik.

Einflüsse und Vorschriften von Iñstitutionen im Schiffbau

See-Berufsgenossenscbalt - Klassifikationsgesellsthaftefl SctUffssicherheltsvertrag - Schiffsverrnessung

FreibOrdvOrschrif-ten - Hauptibmessüngenlrormparameter und DarsteUung dee Schiffes.

Hydrostatische Berechnungen im Schifibau

Flachenberechnung - Momentberethnullg - F'lathentrSgheitsmomente.

Schwinirnfahigkeit und Grundlagen der Stabilitht

Gleichgewithtsbedingungen - Querstabilitlit.

quer- mid Langsstabilität

Die Hebetarmkurve - StabilitAtsbilan.z - KrSngende Momente - Einflul3 des Séegangs auf die Stabilität - LangsstablUtSt/ Trimm - SchwimmfShigkeit und StabffltSt des besthSdigten Schiffes.

Stabilitätskontrolle

KrangungsverSuch - Rolizeitmessung.

-Schiffswiderstand

Ubliche Hande1sschie - Reibungawiderstand - Restwiderstafld - Mocieliversuche.

Widerstandskorrekturen für Probefahrt und Dienstbetrieb

Schiffsantriebsorgáne

Vortriebsarten - Der Propeller - Strahitheorie - Geometrie des Propellers - Propelierdiagramme - Propeflerkavitation.

Zusammenwirken von Schifi und Propeller

Zusammenwirken von Motor-und Propeller

Gesichtspunkte für die Wahi der Antrlebsmascthine, der PropellergrSfle and der Wellendrehzahl - Drehmoment- und Dreh-zahisdiwankungen beim Zusammenwirken von Motor and Propeller - Gleichgewleht zwlschen AntiebsdrehmOment und Pro-peUerdrehmoment - Die Robinson-Kennielder - Umsteuern im Robinson-Kennteld - Bdschleunigen mit aufgeladenem

Die-selmotor - Betrieb mit Verstelipropeller.

-Manövrieren

Bewegungsgleidiungen im ortsfesten bewegten. Bezugssystem Linealisierungen Gierstabilitat MandvrierfShigkeit -Manbvrlerversuche - Drehkreisfahrt - Drehkreisberechnung - Steuerorgane - Querstrahlruder - Ruder.

Bau'weise und Verbände des Schiffskörpers

Festigkeit des Schiffskörpers

Beanspruchungsarten - LSngsfestigkeit - Querfestigkeit - Torslonsfestigkeit - -Detai]iestigkeit - Werkstofte.

Berechnüngsmethoden von Schiffsschwingungen

LI Aflrn..nsa0a tLLt*JS1fl

Der Ladebaum (DerricklSpeedercran&VeUe_Schwingbaum/Haflefl-SchWiflgbaumfStillcken-SchWergUtgeSchirr) - KrSne - Ftas-sige Ladungen - Feste LadungsgUter - Roll-onJRoU-off-Verkehr - Umsdilagszeiten.

Alte/Matthiessen

Schllfbau

kurzgefaflt

Schiffstechnik Bd. 28-1981

Für Studium und PraxIs

168 Seiten mit 127 Abbildungen (Schiffsrissen, Kur-yen) und Rechenbeispielen. Preis -kárt. DM- 29,.

Sofort lieferbar.

Em

Leitfadeñ über alle Gebiete des S±iffbaus in

kurzer und auth für den Nichtschiffbauer

verständ-licher und übersichtverständ-licher Darstellung. Das Buch

schliel3t eine bis heute noch vorhandene Lucke 'im Fachschrifttum und wendet sich vor aflem an

Nautiker

Schiffstechñiker

Schiffsbetriebstechniker

und Zulieferer

aus -den Bereichen der maritimen Industrie.

B e s t eli s c h e i n an Scbiffahrts-Verlag ,,Hansa" C. Schroedter GrnbH & Co. KG, Postfach 1103 29, .2000 Hamburg 11

Ich/Wir bestelle(n) ... Exemplar(e) ,,SthIftbau kurzgefaBt" zum Einzelpreis van DM 29, inkl. MWSt.

zuzuglich Versandkosten.

230

-Name/Firma Datum

"hyp.rbo1ic domain

(1/3.cq2<11

Global view.

Fig. 3 Chat'ige of character of the differential equation (3.1)

from hyperbolic to elliptic for the case of the Inuid S 201.

hyp.rbohC.domain 1/3 .4 q2 Explanation of symbols and notation: Stream tine -1).

-L2

-10 F..P 10

- 08

0& 04 0.2 1.(hPtic" domain q2 1/3,no waves) 0.005 .Water line huLl surface 1nuud 5 201

WatrIrie

I ; 4

"elliptIc" dcmain.(q

.1) -1.005 -1.000

\

(Stagnation poü* at -1.0001;end ofgenerating source distribution _{:x= -.1 ).}

231 - _{Schiffsteehnik Bd. 28 1981}

If, along a ray a

const, we consider a wave system

ff12

(x,y,O) =

A() exp-k Rcos(a-y)}exp{kRccs(c-y)}

dy

(4.3)

-ir / 2

it is known that with increasing R all waves extinguish.each other through

interference except those, for which the "phase" kRcos(ct-y) is stationary with

respect to y, i.e. for which we have

I

dk

d(kcos(a-y.))

_{= 'k cos(a-y)}

_{(tan(cz.-1) +}

_{)- 0}

d y

We have

1

dk

-

₂

For q2=1, (4.4) is satisfied if tan(cL-1) =-2tan y,i.e. if

tan *

-.

The above square root

CL° = arctan (I/V') =190 28!

K

Note that in the analogous case

of

waves over water of depthh, but q2equal 1,

we have the dispersion relation

cos2y

from the conventional

Schiffstechnik Ba. 28-1981

K sin2y. =

1±11. -8 tàna)/4 cotan a..

''

(4.6)

is real if

tan al does not exceed tan

a°.

K

= k.h tanh (kh) /kh

2

2 sin 2y

(1+2 cosZy)q_I

linearized free surface condition and thjs

I

dk

sin2y

'k dy

_{k.h/cosh2(kh)- cos2y}

(4.4)

(4.5)

(4.7)

(4.8)'

Let us confine ourselves note tà the range -ir/2<a<ir/2, -ir12<y< .ir/2. All relations

derived will remain valid with a and/or y shifted by ±ii

In fig

4 we have plot

ted lines of stationary y against a for different va1ue

of q2. The case of

negative y corresponds to a change in sign of a and vice versa

For Ia-i I>/2 in this range, (.3.4) can be met only fork negative, thus

this

domain, which includes the range q2<1/3, is not relevant to our investigation

We find that

For l/3<q2<O.7

and

_{.a<CLN=O.5arc}

cos(1/q2-2), there is one

stationary va

in the range -rr/2<y<O

If O.7<q2<1

,

there is some cLK> CLK such that only for CL<CLK we find statior

ary values-(=tào), located in

the range -ir/2<y<O

For q2>1, for any a there is one staionary value in lherange

O<y<ir/2. for

a positive.. Again we fifld CLK

now

such that we have two more

-I.o -30. 0 30 20 II

-

-If _ -10

-;S I_._' 70/

cu1'

/ / /

_/

,).y

_;

I ' .1

-,

F. C74d" :1 a domut'd ddpaij II I !;Lr,bl5rn i irn tvn'O .2 p

'

' S S ',' '...

'

'

5'

Fig.4: Stationary phase angle

'

in dependence 1mm

coordinate angle ,,

(both measured against doih1hndv

flow) and 1mm q'. 10,.0 -3D -4,fl -50 70 -80 11) 10 In S 0 10.,0 ( .'.fi1s'r.

along lIMPS aliow

t,I' 'a luc of P1jj

/v)i )

30 20 125 21 10 '.0 50 60 .70 80 90 I 10 1.025 Fig.5 :

Stationary phase angle

stationary values in the range -ir/2<y<O for any a with O<a<rr/2.

For the sake of comparison, we have plotted in fig. 5 a

corresponding diagram for

the case of ship

waves on finite depth h for the special case q2 = I with the

parameter Fh standing for the dekth Froude number, Fh =

U/Ik.

The above results are of relevance to Ursell's

181

study on wave propagation on

a non-uniform flow based on the hypothesis that the

phase velocity c for a given

wave numberk is Uq cosy, i.e. that it must balance the

omponent of the

double-body flow normal to the wave front. But from (3.5) we find

c

=

Uq cosy = v'g(l + k)/k

(4.9)

rather- than the con,entiona1 c =

,7j7j

_{underlying Ursell's work. With c}

depend-associated

ing on k through (4.9), the

group velocity

(11)

with

c*2

U2(1-q2)/2 as defined earlier. Note that (4.11) is in general different

from c/2 and even may attain negative values 1

In fig. 6 the variation of the ratio of group

velocity cgto phase velocity

c

Uqcos'y against y is shown in dependence on

q2

.

The

y value

where Cg changes

sign is just the zero point a = 0 of fig. 4, as the condition

cos2y =1/2 - 1/2 q2

inserted into the stationary phase condition yields

in this case. For

q2-2q2sin2y

tan(y-a) =

tan(y-c

Schiffstechnik Bd 28 1981

_/

(1f!2cos2y)q2_1

- 1) <X<

(4.13) degenerates to

Thus we find y = ii/4 for c0. Accordingly,

whatever the value of q2

C

will

remain positive for

y<rr/4.

-If we accept a principle that waves cannOt develop in a region where the group

velocity has a component opposite to the direction of wave front propagation,

it must be concluded that those qua4rants of fig. 4 for which a and. y have

equal

sign must be excluded from our investigation. This would imply two enforcements

of the restrictions (3.7) viz:

(3q2 - 1)

2

both 1/q2 - 2

and

-1/q2

<cos2y <

I

tan 1

(4.12)

234

-and

(4.14)

comes out as

C 2

1_2

(1+

q 2

Cg =

C + k

=

c.(1

k tani /

)

(4.10)

=:

_{+ _----}

)

(4.11)

2q2cos2y

2 sin2y

_{(4. 13)}

2 + cos2y

D. I C. 2 ..._q

O,3'.

U,,'.

0.5

_f

0,6

0,7. 0.8

_{I P}

0,9

N

-1.0

2.0

20 30 '0 50 60\ 70 80 90

Fi.g.6: Ratio of group velocity _{to phase velocity cfr different q2.}

thic domain

is excluded

under restrictio

bh. (4.14)

3. 0 12 1.0 1' 1 4.0 Fig. 7

Rrnge of admitted w.ive lengths in

accord iith (3.7)

3n4 (4 14)

In his equation (4.6), Ursell defines a family of curved rays through the dif:feren.tial equation + (3uu + 2vv x Schiffstechnik Bd. 28-1981

c

sinO-v

dy g dx a cos 0 - u g (4. 15)

In our notation, the r.h.s. is tan(+ct), where a is the angle of stationary phase shown in fig. 4 for y (or A) given, along the line q2 = I marked as "Kelvin's line". As a changes sign there only together with y , we may conclude that under

such approach rays with outward orientation against double-body flow at their origin (i.e. dn the ship's water line) must always cross stream lines with

outward orientation, as no change of sign of a occurs'.

The situation may becOme different if we use our generalized stationary phase relation (4.12) for determination of a, i..e. if we take account of the dependence on q2 . As may be observed in fig. 4, for y large enough, rays may cross with

opposite (= inward) inclination where qZ>1. In particular, they must cross the

ship's water line thn. As this cannot occur in reality, we may speculate if rays

just terminate (invaiiidating Ursell's basic assumption of a purely solenoidal wave number vector fileld) or at least give rise to sudden changes of the side=

wise flow component s measured by Kitazawa, Inui and Kajitani 91 in a certain

region sidewise but losë to the hUll.

It is well known that any ray-tracing analysis based on the.conventional dispers=

ion relation A= 2irq'/k0.cos2y embodies fundamental difficulties near the s'tag=

nation points with q2 approaching zero (see e.g. Keller _51)..

It may be observed that the asymptotic direction of the line q2 = 1 in fig. 3

must have an angle equal arc tan

(l/lY)

against the y-axis. This follows from the fact that far awy any double body flow approaches that of a doublet, and

with squares of disturbance velOcities neglected', only the sign of u + U will become relevant.

It is hoped that the relations derived above may help in finding mathematical models which finally can yield criteria for incipience of flow irregularities

at the 'free surface like wave breaking and so-called surf ace shock waves as

experimentally observed by Inui, Kajitanj and Miyata 113 for extreme ship forms,

where the variation of q2 may be significant over wide areas. For the Inuid S 201

the maximum value of q2 is around I I However, for a vertical circular cylinder q2 rises up to 4.0

5. Alternative 3-D slow ship bondary

conditions on the plaz= 0

Just as in higher-order thin-ship theory, it seems to be unavoidable that

arith-metical and conceptual errors creep in when an analysis is established the first

time. Thus the author only, recently detected some minor inaccuracies in his analysis with regard to terms linear in and The expression

as derived in our result (2.3) should repace the term

y

+ uv ) + (3vv + 2uu + vu in (12) of ₁₃₁

y x y y x _y

-as will be shown in the sequel under an independent derivation of eq. (2.3). Certainly, this correction does not affect the conclusions drawn, but it is essential to make the exponential variation of wave amplitude as described

earlier accessible to.physical interpretation, as thus the dependence.of this

variation on the pressure gradient is displayed rather than on double-body flow direction.

One efficient way of error control 'is 'obviously the comparison with parallel

work of other authors. We have already investigated the connection of _our

boundary condition (2.. 3) with analyses of Newman, of Baba, and of Inui

-Kajitani. Let us turfl now to Dawson

I 1OJ :'

In his eq. (12), Dawson deliberately satisfies the free surface condition (2.1) on the plane z = 0 rather than on a wavy surface, making use of w = 0, u 0

and v = 0 on z = 0 and neglecting square terms in q. Thus two terms of order 0(U3) and order 0(U5) resp. resulting from a Taylor expansion in the vertical direction are omitted. The first one is gCW, a significant component of Baba's

Dr(x,y). The second one is gtq. This term is fundamental in its _{role of}

changing the character of the horizontal differential opera'tor from parabolic

everywhere to elliptic for q2<1/3 and q2<I, hyperbolic for 1/3<q2<1. For the sake of record we should acknowledge here that Takekuma Ilj _{in 1972 already}

detected such ambivalence of the operator and found the importance of the

line q21 for the limiting case of zero Froude number.

-Dawson rightly observed that the geometry of double bod,y flow stream lines is more pertinent to express an essential part of 'the boundary condition than the

c'artesian coordinate ss tern the particle just does not feel 'magnitude and

direction of the undisturbed flow far away. However, at least in the 3-D case,

there is 'another geometry of equal importance, namely that of the double-body flow pressure gradients.

It may be mentioned in this connection that 'our boundary condition (2.3) under

use of the identity

U2q2 = +

ss xx

2

2uv

+v

xy _,yy

rxx -

ryy -5qqU2

(5.1)

where /s = 1/qU (uaIx + v/By) st'ands for differentiation 'along the

direct-ion of double-body flow, can be cast into t'he compact form

U2(q/s)24+g4

_{= g{((u+))}

+

ryy

' (5.2)

The r.h.s. of (5.2) is, the divergence (independent from choice of coordinate axes) of some parallel flow with 'components

_{vy} between the surface}

z=r(x,y) and the plane z = 0 in the sense of long wave 'shallow water

hydro-dynamics.

That his final boundary condition for the plane z = 0 could be expressed

through derivatives in streamline direction only,' seems to stem from his terse analysis prior to his eq. (5.2), indicating that a function constant 'along the

gradient of double body pressure must be 'so along streamlines and vice versa.

In a pioneering study Dagan 12j investigates a formal naive expansion in terms of odd powers of U like

". U1 + U33 +

+ ... , finding th'at each of the

ensuing boundary value problems could be solved with Rankine sources on z = 0

without waves.

We showed'earljer that waviness can obviously be achieved,, starting from

double-body flow, by postulating = 0(U3) and satisfying (2.2) up to er'ror 0(U5)

rather than'requiring' that _{should be proportional to any power of U. (The}

second condition is just enough to eliminate square terms in from (2.2)). We shall show that we could equivalently 'have postulated invariance of wave

resistance up to errors of order o(U°).

The second term of the above naive expansion may however still provide a

meaningful correction to the first one, i.e. to the double-body flow. It' is the solution to ="Dr(x,y)/U3 on z = 0. Gadd 141 usedthe complete naive

potential plus'ã wave correction in his Ratüciné spurce method. Keller also includes 3 before starting ray tracing. And Dagan incorporates into his boundary condition with U1 = U3ct3 , v = (u +2uu1)

+ (v+

R

p ff

({(2

S Schiffstéchnik.Bd. 28 1981

'Iyy +

(2uv+uv3+uiv) + uwi4 +

WIyz +

= 0

Uc3, w1

y z )/2}n - dS

S

Xfl

238

-on z = 0

for which we feel unable to deduce a relation to our analysis.

6. Deduction' of the z=0 bouidary condition from a postulate pf resist4nce invariance

+

pgfc2/2 dy

C

5.3)

There is general consensus nowadays that the leading term of an expansion of

wave resistance in poers of U (provided there is convergence of such expansion)

should be small of order U6 at least - as predicted by thin ship theory - if not as concluded by Baba 61. If then a mathematical model for the wave flow

allows evaluation of wave resistance only in terms of far field flux of momentum (e.g. in terms of a Kochin function squared), the acceptance of such a model shou depend on a ëonfirmatión that this quantity differs from a resistance obtained from integrating pressure over the hull at most by a term smaller than 0(U6) as

long as such flow model has not displayed explicit meiits in the prediction of

near field flow We shall therefore investigate now invariance of wave resistance

under change of control surfaces. It is well known that the linearized'free surface condition of thin ship theory together with Laplace's equation in the lower halfspace provides invariance of MicheIl's wave resistance, whereas the second order free surface condition serves this purpose for the second order wave resistance, as demonstrated in an appendix to an investigation by Eggers and

Gamst 1151.

Let again (x,y,z) stand for the "exact" velocity potential. For an open control

surface S enclosing the ship from below and intersecting the free surface along

a line C the wave resistance can be expressed as

(6.1)

elevation with regard to

free surface. For reason a

in

as shown by Newman ₁₇₁ . Here (x,y) stands for the, wave some reference level z 0 which may be the undisturbed

of simplicity let us assume that the surface S cuts this plane vertically closed contour C.

The 'oriehtation of the normal vector and of dy alongC should be such that

both dy and n. are positive at the ship's stern, i.e. n should be directed into the ship, and the line integral has to be taken in clockwise sense for our case

-We shall now, first simplify the expression for R and then search for simpli= fications for which may allow a direct determination and still preserve the

invariance of R under variation of cbntrol surfaces up to terms 0(U6).

On the free surface, we have Bernoulli's equation

pgz = pgc(x,'y) -p(4S2 * 2

U2)/2 (6.2)

Let us decompose as ₌

r where r again stands for the double-body

potential, which is 0(U). Let us assume that = 0(U3). Then,

if

z = Ois the

plane of symmetry, we have there = °'rxz = 0 and _{ryz = 0 and hence}

= O(IJ), 4 = 0(U3) and OU), and R may be approximated up to

0(U6)

through = +

- U)/2}nx

S0 + pgf/2 dy

'xx

dy - dx)

PfCO(.2 +

+ '-' U)/2 dy C₀ C 0 (6.3) - 239 - Schiffstechnik Bd. 28-1981

with -(2(x,y,O) + 2(x,y,O) + 2(x,y,O) - U2)/2g (6.4)

where in the second line of (6.3) all flow components should be evaluated for z = 0. is the part of S below the surface z = O properly extended up to

C0 there. is negative..

It may be observed here that the last term of (6.3) is just minus tie the single integral ofthe first line. This means that the "static pressure effect"

from the waviness

of

the free surface is just converted in

Gig-n if

we incorporate t1e action

of

dynnic pressure I

Let us consider now two different control surfaces S1 and let ₂ enclose S.1, and., let C1 and G'2be the coresponding curves cut outin the plane z = 0.

The differenceA,R btween

eva'uated over 2 and R evaluated over S1 may then be decomposed as R

_4sR+cR

, with.

p'( If

If ){!;_U2flx_xn. dS

(6.5)' .2 1 -and cR* ---pg( f ' f ) dy p( y - y dx) (6'!6) C2 C1

_C2C1

-From the secOnd identity of Green, applied to the pair of potentials.

we find

-*

If

j'y + pfff(

+ + Y'dxciydz (6.7)

where the first integration is over the surface Sj 2 between the closed curves

C2 and C1 in the plane z = 0, the second is over tie volume enclosed by S2

and _1,2

invoking Stokes' theorem, we find

CR* _{= _p}

ff{(goo

dxdy

(6.8)

Introducing D*(x,y) we then have

D+ 2

4

r

rxx

(o)-,

with1

= + + , using (6.4)

yy.

xx yy zz + 2 C (d +t

ryy

rxx yy

Schiffstechnik Bd. 28 --4 1981 - 240 -* =,

-pffe2

D*(x,y)) _{dx dy} +

fff

dxdydz (6.9)

12

V12

We shall now make use of our above decomposition =cI + and look for.an approximation to satisfying the condition that R should still be invariant up to terms of order o(U6) if is replaced by c Inserting into (6 9), as the choice of and S2 should be arbitrary, we have to require that the integrands

of (6.9) are o(U6).

With Dr r'-' + (trv)

, u and v as x and y double-body flow components for

z = 0, we find X y

D*

= D+

_rxx

_ryy

-u/g. (u k

v) - v/ge (u

+

Vy)y +

o(U5)

(6.10)

- (u2q' -4- 2uvq

+ v2)/g +

o(U5)

With regard to the triple integral of (6.9) we have to require that

trx

I'X)XX

4yy +

o(tJ6)

6.11)

since for _r Laplace's equation íè already satisfied exactly in the domain under consideration. If we can now find a potential which is 0(U3), satisfies 0

and

(u2 -

+

_ryy

+ g

=gD(x,y),

(6.12)

then obviously D, is

0(U5)

and hence (rx + -

D) is

(U6) as requir ed.

7. On the change of character for the boundary condition in the -plane z=O

We should always keep in mind that eq. (6.1.2) is a boundary condition to a

boundary value problem which is formulated for the lower half space z< 0, and

that this boundary condition contains differential operations regarding three

variables In modern terminology, this boundary value problem, if connected with Laplace's field equation 0, is specified as elliptic-, if the associat=

ed boundary condjtion (6.12) has constant coefficients and if a certain discre=

minant D, namely the product of coefficients (i2-g) of and (v2-gc) of 4 minus half the factor of the mixed derivative squared is positiv. For this yy

case, a certain degree of. regularity can be guaranteed for the solution of such problem (cf.App B of 15). In-analogy to classical analysis, it may be tempting

to distinguish between-the domains in the plane z = 0 where

= IJ/4 '(1--4q+-3q) =3U/4 (q2-1/3)(q2-1) (7.1)

is negative, zero, or positive and name these domains hyperbolic, parabolic, and elliptic. We should however emphasize here that very little is known about the

implications of such a distinction with regard to the local character of the solutions to our problem. At least, there is no evidence that trespassing the parabolic border lines q2 = I and q2 = 1/3 of these domains must be connected with whatever kind of irregularity or discontinuity so far. For the hyperbolic range 1/3<q2<I, One may, nevertheless, seek for -analogies with--supersonic flow

problems. If we suppress the only gravity-dependent term g

z (for high Froude

number(?)), we may exclude the third dimension in (6.12).*2

With c-

_{= 1r}

=U/(1_qZ)/2 in this range, dividing by - c , we obtain

(i-_

_{xx1_}

_{- 2k02I(1-q2,)}

=

D(xy)/c'

_{((1-q2)qusin8).}/ (1--q2)}

(7.2)

One may even define characteri.tic curves here throug.h the ordinary _differential

equations

-* *2

dy -uv ± c/U q - c

= tan (8±aM) in our notation

dx

_{2_u2}

with = arc sin (c*/Uq), which degenerates to

= tan 8 for q2 = I, i.e. c* = 0

dy. U

-2

= - _{= -- cotan 8 for q = = 1/3}

and to

This shows that such characteristic lines _{are orthogonal to the double-body flow}

near the stagnation points, but they merge tangentially into the stream lines

at the lines q = .

- 241 - Schiffstechnik Ed. 28-1981

8. Speculations on alternative (non-linar) first order flow models

(Preiswerk's 2-D shallow water waves in particular)

The assumption of potential flow thay be inadequate near the ship if we in-.

tend to admit flow irregularities there, we should epn expect nun-zero contributions to the free surface integral part ofR pffu(w-D (x.,y))dxdy from lines of flow discontinuity where dissipation of energy due .to wave breaking ma occur. In particular; our basic assumption that all deviations

from double-body flow are small of order 0(U3) should be weakened in such

a domain.

In the spirit of 141 , one may rather seek for a basic flow with components

u, v, w

subject to the following conditions: .

U.

+ V.

+ W = 0 for

x _y z

Substituting h for

Schiffstechnik Bd 28 1981

z < 0 outside the volume displaced by the hip

Un + vn + wn = 0 on the hull surface S for z < 0

x

y.

z -. o

u-U, v, w

- ₀ for x2+y2+z

D*(x,y) _{0 for z = (8.4.a}

_{w =}

_{0 for z = 0}

{1 -

}u + l

-

}v

- {u

+ V

} =

gh x gh y gh y

.x

- 242

-we may observe that from (8.4.a)., (8.4.b) -we obtain

(8.1)

(8.2)

(8.3)

(8.4.b)

with D*(x,y) =

(7u)

+ (0v)

= (U2-u2-v2--w2)/2g as in (6.9). (This is obviously stronger dan vanising of the integrand oftR in (6.9) ! ).

This basic flow could replace the double-body flow componentsin our proceed-ing analysis for detrminproceed-ing the gravity-dependent wavy flow, say in our

partial d.é. (6.12) which then would come out

homogeneous (i.e.

with zero r.h.s.).

-It is well known _7! that (as Neumann problem overdetermined through (,8.4.a))

(8 1-4) does not admit a potential flow solution, but various solutions with

non-zero vorticity in certain domains may exist.. The character of such

vor-tex distribution may e.g. depend on Reynolds number from the presence of a

free-surface shear-flow and/or of the ship's boundary layer 1161. It seems

to be permissible, however, to assume zero vertical vorticity such that in

any horizontal plane. u and V represe1t some potential flog.

One should observe that all essential steps leading to (6.8) and (6.12)

could be repeated under assumption of non-potential flow, though the global application of BernOulli's law requires that any curl vector must be in flow

ditection, and some lift force acting on such curl cannot be disregarded.

(8.5)

This identical with Preiswerk's 1171 equation (14) for- stead.y 2-D shallow

water flow from an undisturbed level h (corresponding to U2/2g in our case).

Introducing the 2-D potential , Preiswerk discusses the

non-linear

equation

With q2 = (u2 + dx

_{c*2_u2}

Wehausen/Laitone: u our notation: u Prejswerk: u

_!.)

.

+ (j_i)

-2 )

gh X2 _g yy gh xy = ( + and c =

this d.e. is hyperbolic with characteristic lines given through

in strict analogy to (7.3) in the domain where 1/3 < q2 < I

On _{may observe, however, that the frontiers of this domain, i.e. the lines}

q2 = _{1/3 where the stream velocity coincides with the shallow water wave}

velocity,and the, lines q2 = I (i.e. h = 0) haire to be determined

nwnerical-ly _{This can be achieved however through solving the equivalent li.near}

par-tial differenpar-tial equation

v2 U2 Ut

+ (1 _- + 2 _{xt, =} 0 (8.8)

for the function X(U,V) obtained through the Legendre contact transformation

x = xu + yv- as a function of the velocity components u, V in the

hodo-graph plane The characteristics of (8 8) may be represented in closed form

after introduction of polar coordinates q , through u

U q cos,

v = U q sine , transforming (8.8) tc

Xqq X

(Uq2

I)

*

Xq

! (Uq

1) } = 0 (8.9)

as = ± /3 ärcsin(3q2-2) + arcsin(1/q2-2)}/2 constant' . (8.10) i.e. as epicycloids between the concentric, circles' q2 = I and q2 = 1/3. The transformation to the x-y plane is perforthed through.x _{= y x'}

An extensive discussion of above analysis can be found in Preiswerk's hesis 17J and within the broad exposition given by Weháusen and Laitone 18

pp 682-695. For easy cross reference, we display the intet-relation between

the symbols used.:

The "Froude.number" F is the inverse of our sin - Preiswerk's c is v'3 times otir q

/

- 243 - _{Schiffstechnik}

_{Bd. 28 1981}

(8.6) (8.7) h c _c fl F

V Uq U2/2g c U//3 -1/3k Uq/c*

So far, there is no indicatiOn how Preiswerk's 2-D flow model could be

matched to the double-body flow which is dominant on the ship's hull and

in the far field. Nevertheless, it may provide essential qualitative

in-formation on the geometry of the domain where dispersive waves can be

expected and on lines of discontinuity The investigations under I 19,201

seem to be concerned with a simplified variant of 1171 with h assumed

constant.-Acknowledgements;.

The author's thanks go toMr. Huang.Ding]5ing from DalianTechnicallnsti

tute for support in editing,to Dr A Aldoan for meticulous proofreading and to Frau L.v.Maydell for dLli'gence and patience shown typing the papet.

References: /

Longuet-Higgins,M.S. and Stewart, R.W.: The changes in amplitude of

short gravity waves on steady non-uniform currents J Fluid Mech 10 (1961), pp. 529-549.

Longuet-Higgins, M.S. and Stewart, R.W.: Changes in the form of short

gravity waves on' long waves and on tidal currents. J. Flui,d Mech. 8

566-583.

Low speed wave resistance theor making use of strained Research Report (1978), Study on non-linear effect in

pp. 7-31, discussion by Eggers, K., pp. 33-38.

Inul, T.; Kajitani, H.: A study on loca1non-linear free surface effects in ship waves and wave resistance. Schiffstechnik 24 ('1977),

pp. 177-213.

Keller, J.: The theory of. ship waves and the class of streamlined hips. J. Fluid Mech. 92 (197.9), pp. 465-488.

Baba, E.: Wave resistance o,f ships in low speed. Mitsubishi Technical' Bulletin No. 109, Aug. (.1976) (20 pages).

Newman, J.N.: Linearized Wave resistance theory. Proc. mt. Seminar

on Wave Resistnce Theory Tokyo (1976), pp. 31-49, disc. 393-401.

Ursell, F: Steady wave patterns, on a' non-uniform steady fluid flow. J. Fluid Mech. 9 (1960), pp. 333-346.

Kitazawa,, T'., Inui, T. and H. Kajitani: Velocity field measurements applied, to analysis of ship wave making singularities. Proc. 10th

Symposium on Naval Hydrodynamics Cambridge (1974) pp 549-561,

disc. 562-569.

10] Dawson,C.W.: A computer method for solving ship wave problems.

Proc. Second Intern. Conf'. on Numerical Ship Hydrodynamics, Berkeley (1977), pp. 3O-38. ' I'51 161 191 F Ii I (1960), pp. Yamazaki, R. coordinates. ship waves, Takeku±na, K.: J. Soc. of Nay Schiffstechnik Bd. 28 .1981

Study on the non-linear free surface problem around bow. al Arch. of Japan 139 (1972), pp. 1-9.

-12! Dagan, C.: Non-linear ship wave theory. Proc. 9th Symposium on Naval

Hydrodynamics, Paris (1972),

_pp.

1697-1737. '31,

wave resistance of wide-beam ships. J.Soc. Naval Arch. of Japan 146

Tnui, T.; Kajitani, H.; Miyata, H. -et alii: Non-linear properties of

(1979), pp. 19-27.

141 Gadd., G.E.,: WaVe theory applied to practical hull forms. Proc. Intern. Seminar on Wave Resistance Theory Tokyo (1976), _pp 149-158, disc

pp.

426-429.

1151 Eggers, K. and Gamst, A.: An Evaluation of mapping procedures for the stationary ship wave problem Schiffstechnik 26 (1979), pp 125-168,

disc. pp, 169-170.

1161 aba, E.: Some ftee-surf ace phenomena around ships to be challenged

by numerical analysis. Proceedings of 3. Tnt. Conf. on Numerical

Ship Hydrodynamics, Paris (1981).

171 Preiswerk, E.: Anwendung gasdynamischer Methoden auf

Wasserströmun-gen mit freier Oberfläche. Mitt.. Inst. f. Aerodynamik, EidWasserströmun-gen. Techn.

- Hochscl-iule Zurich, No. _7., 130 pages (1938).

1181 Wehausen, J..V. and E.V. Laitone: Surface waves. Encyclopaedia of Physics, Vol. IX, Springer (1960)..

['91

Miyata, H.; Inui, T. and H. Kajitani: Free surface shock waves

around ships and their effects on ship resistance. J. Soc. Nay. Arch.

of Japan i7 (1980) pp. 1-9.

2 Takäh'ashi, T.; }ajitani, H.; Miyata, H. and K. Nakato: Characteristics of free surface shock waves around wedge models. J. Soc. Nay. A±ch. of. Japan 148 (1980) pp 1-9).

D

i, S C U.S S i_Ofl

Prof.- Hideaki Miyata, Department of Naval Architecture, University of Tokyo

ThE free surface waves caused by ships are very complicated, and it is clear

that classical linear wave making theories cannot adequately cope with the actual phenomena. The author's work is vety challenging and will promote. the development of new wave making theories.

The most outstanding feature of waves is observed in the neighborhood of

ships and the experimental investigations at our experimental tank have clarified the characteristics of the waves

_{I II.}

They are nonlinear wave making phenomena called free surface shock wave Recently Kayo and Takekun'a

21 claim that nonlinear free surface phenomena around bow are dependent on Reynolds number and propose a secondary flow model. This argument seems to be misleading, because the essence of the phenomena is wave making ruled by Froude number although. waves will receive secondary influences by disturb-ances. We-have experimentally demonstrated that the nonlinear waves around bows of wedge models are ruled by Froude number based on draft 31. A new experimental result that supports thi.s concept is shown in Fig.1. The

observed wave front lines of the fOremost free surface shock waves around wedge models are compared at two draft conditions. The shape of wave front

lines experiences considerable change by the increase of advance speed! and this change is ruled by Froude number based on dtaft (Fd). The wave f:ront lines of the two draft conditions at the same Fd accord well with each other, although the advance speed is quite different.. The effect of Reynolds number should be treated after wave making is thoroughly clarified.

The nonlinear governing equation similar to author's equation (3.1) first appeared in the pioneering work byTakekuma (author's reference I11).

However., his work h-as some qUestionable points. Firstly the derivation of the terms that consequently determine the critical velocity (which coincide with author's term g and so on in equation (3.1) is arbitrary.

Secondly he assumes te ines on which determinant is zero to be the lines

of wave crests.

The terms with must be carefully derived because they play a

signifi-cant role. We have utilized the equation(8.6) by introducing a new concept of equivalent shallow water de?th for the practical purpose of estimating

the shape of wave front lines 3.. The critical speed is independent of, advance speed and the wave front lines estimated by -the method of charac-teristics can well explain the variation of the angle of wave fronts (Figs. 19 to 21 of 3! and Fig.31 of 14!. Our practical method of estimating wave front lines was applied to the procedure of hull form improvement and

succeeded in reducing wave resistane. The author's critical speed is

dependent on advance speed, i.e., C increases with the increase of advance -speed. I am not sure whether the wave front lines obtained from equations

(8.7) and (8.8) can explain the actual wave front lines..

The most outstanding feature of nonlinear waves is the generation of lines

Of discontinuity as hown in Fig.2. Fluid velocities undergo sudden dis-continuous change an energy loss occurs at the discontinuity. Therefore. the numerical approach

_Is!

will be indespensable for the nonlinear wave - problem although the author's analytical approach is very helpful.

Ii!

Miyata, H.:

Characteristics

of Nonlinear Waves in the Near-Field of Ships and their Effects on Resistance, Proc. of 13th Symposium on Naval Hydrodynamics (1980)

-121 Kayo, Y. and Takekuma, K.: On the Free-Surface Shear Flow Related to

Bow Wave Breaking of Ful.l Ship Models, .J. Soc. Naval Arch. of Japan,

- Vol. 149 (1981)

-!31 Takahashi, M., Kajitani, H., Miyata, H. and Kanai, M.: Characteristics

of Free Surface Shock Waves aroUnd Wedge Models, J. Soc. Naval Arch. of Japan, Vol. 148 (1980)

141 Kawamura, N., Kajitani, H., Miyata, H. and Tsuchiya, Y.: Experimental

Investigation on the Resistance Component -Due to Free Surface Shoc-k Waves on Series Ships, J. Kansai Soc. Naval Arch., Japan, Vol. 179

(1980,)

-5! Miyata, H., Suzuki, A. andKajitani, H.: Numerical Explanation of

Nonlinear Nondispersive Waves around Bow, Proc. 3rd Conf. on. Numerical

- Ship Hydrodynamics (1981)

-d = O.lm d =

0.15m-Fig. 1

Observed wave front lines of foremost free surface shock waves at two draft conditions

(wedge model, a. = 200) u/U

_'-0.5-

v/U

0:

'0.2

tir

0.

A A

AA222

0

9

12

-o

0

xl'

Fig. 2

Disturbance velocity distributions 'añd head loss

near the front of a

free surface shock wave from stern (y/l

0.15, Fn = 0.30) A

ti

-1.0 1.1 1.2

xi'

4

H=g{U2_(U,U)2_V2_W2}4(

0.0000

0

o-0005

A

o

-0.02

0

- K.

0'

00

001.1

12

Prof. Fritz Urseli, F.R.S. Department of Mathematics,

1niversity of Manchester,

England-This is a very interesting paper which deserves detailed study. Unfortunately

I have only had time to consider a few points which occurredto me when-I

coin-pared Professor Eggers's arguments and results with those in my own work. I assumend that the local phase velocity is equal and opposite to the local basic-flow velocity, and also that the distance between neighbouring phase curves is proportional to the local wavelength. From these local conditions the existence

of characteristic curves was then inferred but the constrLiction of the phase

curves required an additional assumption. Professor Eggers's arguments in §3 are local and his equation (3.5) thus resembles my first condition in this respect. (In §3 it is assumed that the waves are not steep; this may nOt be

valid in some regions where q is not near I, e.g. near the stagnation points.) His arguments in §4, however, use the principle of stationary phase and are

non-local. In fact the parameter R is made to tend to infinity, but in this limit operation should not q also be made to tend to 1 ? Thus I am doubtful about

§4 and would prefer some local argument.

I must admit that I shall not find my own results fully convincing, until I can derive them from-a systematic perturbation procedure, such as the one given by

Keller.

Reply to discussion by Prof. Miyata.:

1 think that the wave flow model presented here can simulate several essential features of the phenomena observed by Prof. Miyata and hence may be considered as a partial alternative to his non-dispersive approach. For a thorough

evalua-tion of this aspect, we need more calculaevalua-tions and measurements of q2 for

cal-culation of wave front lines. We should investigate then whether a subdivision of the free surface into a "hyperbolic" and an "elliptic" domain is of practical concern, say in the sense that "shock waves" cannot start in the latter, though

they might find continuation as part of a Kelvin-type wave pattern there ? Let us introduce two types of curves:

I. _Mach lines

Rather formally, we had introduced certain "characteristic curves" through the

differential equation (7.3)

tan(B+aM).

Now, for q2 given,

_M

is the coordinate angle of stationary phase for the

mar-ginal waves of extreme y as v4ualized in fig. 4, for which the phase- 5and the

group velocity coincide with c , for which the wave length tends to zero, and for which we have - y =

¶12.

Hence the wave fronts are

tangential

to the curves defined above, and we should expect changes rather in the

normal

than

the tangential flow components (just as in the analogue case of waves behind a

ship on shallow Water jfl supercritical speed) if waves are expected to

terminà-te along sucha curv.

-Only such characteristic lines which come close to the ship (i.e. the cause of

any disturbance) can be relevant. Let us thus consider the special solution of above d.e. which passes the point ahead of the bow where q2 = 1/3 (hence _M

¶12)

with 8 = 0 and iame it a "Mach line". With q2 increasing, i.e. _M

dimi-nishing, this curve is concave against the bow just as the contour of the

"shock front wave-A" of Prof. Miyata, and like the "front line of, plateau"

referred to in Mon's discussion to Kayo's investigations I21l A Mach line terminates tangentially to the double-body flow where q2 approaches unity.

-0

/ __

I .c_

)

-X

,)(

--

-2..2/3

-?

s:

/ .c

,//' 0iVj

!/'

_{O_f}

'\ \\\.

H

_/

rl

1.c 1.2 2.0 1.2 0. ..0.

Fig.8.

Mach line, Kelvin line ard characteristic curves (eq.(7.3))

_{for a}

vertical circular cylinder with draft equal

to dianieter.

II. Kelvin lines

We have considered certain rays for wave (energy) propagation defined though (4.15)

-dx

where ci ci(y,q7) in anRloy to similar rays considered by Ursell and by Keller

with q2 equal unity everywhere. In the case of Kelvin's wave n,del, we find that all such rays passing through a fixed point are confined within a wedge

< 4,

and that the wave elevation is most pronounced along the " cusp lines"

a -

4 .

Generalising this to our flow model, we define "Kelvin lines's, starting

near the water line where q2'O.7 (for smaller q2,a is not defined) subject to the differential equation

tan(B+aK)

Such lines start with an inclination of against the waterline, in the hy-perbolic domain, not too close to the bow in case of larger entrance angles. They will finally degenerate to straight lines of inclination

4

against the x-axis

far behind the ship.

It is not obvious that for Kelvin lines as well changes in flow should be mainly

in normal direction in the sense of Miyata's "shock condition". Otherwise, they

seem to show the same features as the secondary "shock front wave-B" of Prof.

Miyata.

Note in particular that - as K>M in general Kelvin lines may cross with Mach

lines in accord with the experimental findings

However, I think that Prof. Miyata's investigations demonstrate that we should be prepared to encountei a pronounced (and not necessarily monotonic) dependence of

the wave front geometry on the speed U for certain hull forms. This is not cover-ed by our approach as' long as we take the double-body flow as the starting point.

I am wefl aware of controversial views concerning the question to what degree

such a dependence on sped should be scaled against Froude number or.Reynolds number. For his wedge model, for three pairs of points in the speed-versus-draft

plane., the discusser's results may be interpreted also as simple verification of Froude's law that a geosiin change in hull dimensions by a factor 1.5 may be com-pensated by a proportional increase of jj2, if we assume that the model length was not relevant to these experimental observations close to the bow. In other

words, the experiments do not necessarily imply that draft rather than length is

a characteristic variable for non-geosims.. Moreover, the speed and size range of

Prof. Miyatá's experiments cannot rule out a possible additional dependence on

viscosity, especially for ship forms with relatively blunt bows.

I welcome the discusser's clarifying remarks on the role of a critical speed. In his approach, this quantity (related to his "equivalent depth") gepends on the hull geometry, but not on U, and Obviously not on x and y. Our c is

proportio-nal .to U (this implis that Mach- and Kelvin line geometry depends on the ship

form, but not on U) and it depends on the coordinates within the hyperbolic

domain where it is dfined.

I think that. attempting a discrimination between non-Kelvin dispersive waves and f.ree surface shock wives through experimental observations only is a highly

pro-hibitive task. Hovevr, the evidence of a localized loss of head may serve as a discriminant so long as we do not admit "wave breaking" along the front Mach line

of a dispersive wave system. The impressive experimental data of Prof. Miyata unfortunately are taken from the stern region,, where comnlications from

addit.io-na.l physical sources cannot be excluded beforehand. We shouldhe aware that even experimental fi.nding behind very thin ships, such as the diminishing of the

Kelvin angle observedduring Adachis investigations 1,221, are still waiting for .a rational interprettion !

As the discusser empIasizes the need for a purely numerical approach, I suggest

that he should test Iis procedure on the we.l.l defined case of a vertical

circu-lar cylinder (with dr1aft equal to diameter), for which we have recently measured the wave elevation i.r Several directions around the bow as parts of Dr. Kayo's recent extended experiments in our tank.

Wave resistance research within the last 25 years has experienced decisive

'im-pulses ftgain and again from fundamental investigations at the University of Tokyo. This has enhanced the general conviction that "no effective wave making

theory can be generaed without precise understanding of t'he physical phenomena"

231. Consequently, I feel that there is a need for investigations on a

funda-mental level to provide a physical justification for the fictitious equivalent

depth concept introduced by the discusser, which so far seems to be defined through the experitnental data it has to fit. But I appreciate his investigations, and this discussion in particular, as a most valuable continuation of the

perma-nent stimulus and challenge from Prof. Inüi's school to keep wave resistance research lively.

-Reply to discussion by Prof. Ursell:

I am grateful for Prof. Ursell's comments which partially _{meet my intention to}

initiate a group discussion on a subject where our knowledge so far I feel

-is insufficient and where time -is ripe for _{a synoptical evaluation of the}

pre-sent achievements, i.e. the state of the art. Just as the discusser, I myself do

not find my approach completely convincing, and any suggestion

_for

a modifica-tion or improvement is welcome.

I share Prof. Ursell's intuitive precautions against _{the use of stationary phase}

arguments in the course of a local investigation, However, I invoked this prin-ciple only in order to determine the direction in which _{a wave group might} pro-pagate, and my result (4.15) is precisely equivalent to Prof. Ursell's finding

that "It is noteworthy that the characteristic direction is along _{the resultant}

of

the group velocity, taken normal to the phase curve, and the vejocity of the

basic flow". (See page 339 of 81). It _{seem noteworthy to me that this} equiva-lence is independent from the nature of the underlying dispersion _relation

k = k(y)

But the formal identity between my (4.15) and the discusser's (4.6) _{does not mean} that the curves found through integration àr& the same ! Whereas (4.15) ias con-ceived for curves along which the wave front angle y _{remains constant, Prof.} Ursell's (4.6), on the other hand, is part of a system of two simultaneous

diffe-rential equations, the second of which (i.e. (3.7)) isdetermining _{the variation} of 8 (and hence of y =8- _{) along such curve, acting at the r.h. side of the} first. I sense that with variation of the wave front angle even the wave length

cannot be. considered invariant along, such curve.

Thus, with y given at an initial point, _{we should obtain different curves even} for q2 = 1, thoUgh with the same initial tangent. - Anyhow, if y should remain

restricted to its quadrant, I cannot understand why _{rays with positive initial}

inclination a against the water line can ever approach the hull again, as may

be observed in computations of _{4! and still more drastically under Yim's 24!} recent investigations, where evenmultiple reflection is observed. Note that Keller 15! even excludes the possibility of _{rays emanating alongside the} fore-body, as hc concludes that they would _{penetrate the hull.}

-It is known from many ship wave calculations that the stationary _{phase method} yields fairly good approximations _{to theoretical wave pattern even close to the} cause of disturbance. Nevertheless, I agree that the assumption of short wave length

(say

_{compared to the radius of curvature of the double body flow) is}

indispensible for some of my conclusions. But, whatever _{the value of q2 locally,} it will be sufficient to keep the Froude number _{small. And the process of energy} dissipation near the, free surface (through "wave breaking", through "shocks",

through "wave focusing" or whatever mechanism) which _{we finally hope to clarify,} seems to be related to the short wave range of a ship _{wave spectrum only.}

It was not explicitely assumed that _{waves are not too steep in §3, and even near}

the stagnation points where the wave elevation should rise _{up to. U2/2g, the} amplitude of the wavy flow component in the plane z 0 need not be excessive.

But an inclusion of non-linear components should certainly _{be aimed at.}

Additional references:

121! _{Non, K.: Discussion to "Observation On the free surface phenomenon}

around the bow of a ful,l form". Proc. of the continued workshop on ship'

wave resistance computations, 10-12. October 1980, Izu Shuzenji, Japan,

pp.. 133-135.

1221 Adachi, H.: On some experimental results of a ship with extremely long

parallel middle-body. Reports of the Ship Research Institute 12 (1975),

pp. 1-15.

Sehifistechnik Bd. 28 1981

23! Inui, T., Kajitani, H. an4 Miyata, H.: Experimental investigations on the

wave making in the near field of ships. Journ. Kansai Soc. Nay. Arch. 73

(1979) Pp. 95-107. .

1241 Yim, B.: A ray theory for. nonlinear ship waves and wave resistance. Proc.

3rd tnt. Conf. on Numerical Ship Hydrodynamics, Paris 1981 (pp. 1-4-I to 1-4-10 of preprints). - .