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OCR for page 157
Ship Wave Ray Tracing Including Surface Tension
D. B. Huang
Harbin Shipbuilding Engineering Institute
Harbin, China
K. Eggers
University of Hamburg
Hamburg, Germany
Abstract
The aim of this work is to clarify the validity of slim
wave ray theories at and near the ship's surface. As
previous numerical investigations have led to ambigu-
ities due to a breakdown of the ray analysis near the
bow and stern stagnation points, we shall take care for
the surface tension effect in order to Olden such defi-
ciencies; then the wave length never surpasses a positive
nimlun length which is attained at the boundary of
a finite waveless zone around a stagnation point. It is
found, however, that the ray equations degenerate at
these boundaries, and that rays can be traced into the
far field only if their starting point is selected outside a
finite belt surrounding the waveless zone.
For a class of bi-circula~ prismatic struts of infinite
downward extent we investigate two alternative formu-
lations of the free surface condition and their implica-
tions for the ray geometry. For low speeds we find in
both cases an increase of the Kelvin wave cusp angle clue
to capillarity. We exte~cled tile ray tracing to capillary
waves ahead of a blunt bow.
Introduction
The wave field at a point far away from a ship in
stationary motion is well represented through Kelvin's
pattern, found in a wedge-shaped region, with only a
finite slumber of wave components, given through wave
length, wave front angle and complex amplitude. The
first two are constant; along straight lines (character-
istics) through a hypothetical origin, conceived as the
locus of a point disturbance.
Observat;ions suggest that under local modifications
such a wave model may be adequate even near a ship;
Ursell [1; hence generalized this approach for waves due
to a point disturbance in a slightly non-uniform flow.
Mom a set of physical assumptions, he replaced the
intensity aloud direction of the uniform flow by the lo-
cal components to obtain an analoguous spatially vary-
ing "dispersion relation" between wave angle and wave
number; from a partial differential equation he obtained
"rays" along the resultant of the local flow with a group
157
velocity vector. l'o simplify the problem, Ursell consid-
ered only rays passing through the disturbance, thought
he admitted that his assumptions are questionable
there. Inui and Kajitani t2] used this approach for waves
near a slip's bow, with the " double body flow " as the
basic non-uniform flow.
Keller [34 derived Ursell's results from a more formal
approach, tacitly assuming pertinence and uniform va-
lidity of his ray theory up to the ship's water line; he
even concluded for certain ships that rays must origi-
nate from the double body flow stagnation points only.
Yimi44 evaluated this approach numerically, but due to
zero wave length at these points he had to start ray trac-
ing using values at some distance. For certain rays car-
rying transverse waves he observed that they re-entered
the hull; to avoid this, he introduced some mechanism
of reflection.
Brandsma t5] investigated a class of bi-circular forms
with varying entrance angle. Even with "backshooting"
from downstream, he failed to find rays due to trans-
verse waves originating at the stagnation points; he
therefore concluded that no transverse waves can em-
anate from the bow.We shall demonstrate analytically
that the rate of change of the wave front angle along
the ray together with the change of rater direc-
tan has a factor as inverse distance from the
3tagr~atior~ points. Thus it demands more detai-
led analysis to Claire fy the validity of Xel-
ler's ray theory near the stagnation points.
Through our present investigation we want to clarify
whether the inclusion of surface tension effects can im-
prove the situation at least to the point that ray theory
can give some qualitative information about the wave
pattern geometry in accord with experimental observa-
tions for not so slender ships. We selected the class of
bi-circular struts and thus have even the case of a blunt
bow included.
-
l Otherwise rays could be extended to the domain far
ahead through backward tracing, at least in case of a
submerged disturbance.
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Representative terms from entire chapter:
surface tension
The underlying analysis was presented by Eggers [6],
where two alternative appoaches were considered: (A)'
based on the conventional surface condition (A) of slow
ship theory, supplemented for surface tension following
Maruot7] and (A+)' based on a modified free surface
condition, derived by Eggerst8] from certain invariance
requirements for wave resistance, again supplemented
for capillarity eEects. In both cases we obtain zones
around the stagnation points where no steady waves can
exist; at their boundaries, only waves of minimum wave
speed, with wave front orthogonal to the double body
flow, can occur. If we start rays from these boundaries
rather than from the stagnation points, we apparently
have a well defined initial value problem, even for blunt
bow forms. -
In our computational investigations, we could con-
firm Maruo's experimental finding that capillarity ef-
fects can be significant even if the remodel speed exceeds
the rnirumum wave speed considerably.
However, we found ourselves confronted with some
instability phenomenon. Due to the strong rate of
change of the wave angle along the ray near its origin,
flee wave length re-approached its minimum value after
a short time and the analysis broke down. To find rays
which can be continued into the far field, we lead to se-
lect the starting point outside some "belt of sI~ort-livity"
surrounding the waveless zone.
The neglect of 9(r <'fizz (the second term of a formally
divergent Taylor expansion) leads to the approach (A)'
investigated by Maruo, which was developed from the
"conventional" approach (A) underlying the ray analy-
ses of Keller, Yim and Brandsma. Let us now consider u
and v as slowly-varying (i.e.locally constant) quantities
and let us disregard effects of phase and of amplitude,
as they are of no concern for investigations on ray ge-
ometry. A potential of the form
`~' or e(kZ-is) with 5 = S(x,y) (3)
represents a wave with wave number vector
k IS—{k:, k2 ~—{k cos 8; k sin 8) (4)
if h~ and k2 are also slowly varying. Here ~9 is the angle
of k against tile x-direction, Let us define the speed
ratio q and the flow angle ,0 through u _ Uq cos,l? and
v Uq sin,l3. Then By _ ~ - ~ is the angle of k against
the flow direction. In accord with Brandsma and Yim,
we have selected the orientation of k such that cosy is
non-negative,i.e. k is , opposite to the propagation
of a wave stationary to the ship.-
': u , 1 Id 7rec lion
Derivation Of Dispersion Relation And pow {ancient :)
Ray Equations From Free Surface Conclitions.
For simplicity, we shall restrict ourselves to a 2-D
flow around prismatic struts of infinite vertical exten-
sion. Let us consider a velocity potential of the town
U¢r + Up, where U stands for the far field uniform
flow in the +x direction, U¢r represents the "double
body flow" (unbounded in tile upward z-direction) and
US is the lowest order wavy potential. With u
U¢rr, v— USA, with (r _ (U2 _ u2 —v2~/2g arid
Dr(~,y)—(u~r)2 + overly, ~ for z = 0 has to satisfy
MU
u2(p + 211v~7 + v2('o + 9~(~0 + :(~0 ~
= 29~(r3
"characteristic stripes" (i.e. characteristic curves in the
(~, y, k~, k2 )-space) can be found from the equations
dz F dy F dk~ dk2
dr ~k~; dr ,~k2; dr = -F~ ; d = -Fy (8)
which define a curve parameter ~ Under multiple use
of above relations, considering that
bI(uk~ + vk2) b(`uk~ + vk2)
{tk~ = u; = v (9)
,~k ~ = cg cos8; ,9k ~ = cg sin O (10)
(where cg —d~kc)/dk is in accord with the concept of
group velocity related to Uq cos ~y as phase velocity), we
find
cg =
We obtain
dx
d~r
In a similar way
Fiom (6) we find
and thus
Fy
dk (kc) = dk>/kg + k2g~r + k3~
2k (9 + 29(r + 3k2 ~) = 2 ( 1 + 9(' +2 ) ( 11 )
63F —kc ~ Lc
= = 2
~k~ uk~ + vk2 ~k~ uk~ + vk2
=~ k ,9k (kc-uk:)= k (u-cg cOs§) (12)
dy = 2 (v - cg sin§) (13j
~Z 2C 61Z 9(r (14)
d k1 kc ~ kc
1;, = -— = - 2-
2: dT uk1 + vk2 ~X Uk1 -t Vk2
- 2 <~ (kc - Ukl - Vk2 )
—— —~—9(r—ukl —vk2 )
kc dx c
= 2 ((uu2 +vv=)/2+c(u~ cos §+v~ sin}))~15)
= 2 ((UUy +vvy)/2-Fctuy cos §+vy sin §)) (16)
Then we find the rate of change of the wave angle ~ from
kc2 dd kc2 d k2
= — arctan
2 dr 2 dr k1
2Ck (ki d - k2 dT )
(17)
= _c2 (Fy cos ~ - F~ sin §)/2
= ( (9 (r )y cO s ~ — (9 (r )3' s in §) / 2
+ cu2 sin 2} - cv~ cos 25 ~ 18
where we have used u~ = -vy, uy = v2 for our 2-D basic
Ilow. In a similar way we obtain5
2 dT 2k (k: dT + k2 d 2 )
((9(r )2, cOs8+ (9l~r )y sin~q)/2
- cu~ cos 2cq—cv~ sin 2§ (19)
From equs.~12) and (13) we can easily confirm the gen-
eral result
tan(3 + cx) dY = g ~ (20)
(witI~ c~ defined as the ray angle against the double
body flow) which contains the choice of approach only
through the explicit expression for cg. We may thus re-
call Ursell's observation that the ray direction is along
tI~e resultant of the basic flow a~d the "group ve-
locity" taken along the wave normal vector -k, and
that this does not require asymptotic analysis such as
tl~e principle of statio~ary phase (see discussion tot84~6
~ Cat
—k
7~+)—~
~<
// ~+~
Uq
Fig.2 Ray direction as resultant of basic flow
and cg along direction of -k
(with angle c~ against flow direction).
Restrictions For The Wave Parameters.
From the dispersion relation (6) we have
c2 = g/k +9(r + /ck (21)
5Tlle terms with 9(r (Iriissing under approach (A)')
reflect the statemellt of Longuet-Higgins and Stewart
t11] that short waves superposed a long wave shorten
wileIl clirrlbiIlg, increasiIlg their length again when
descend.illg.
GFor later use, we have introduced in Fig.2 an "ac-
tion transl~ort velocity" cat along tile tangent to the ray
direction.
159
A minimum of c is found at k = >/577 giving
C = Cm at +2~1 - 92 )/p2 = U:( 1 - q2 + 2p2 )/2 (22)
where c'', = .,,~'4'cg is the minimum velocity of capillary-
gravity waves and p—cm/U is a dimensionless pararn-
eter of surface tension. We introduce a dimensionless
wave length A—g/(kU2~; then (5) is equivalent to
q2 COS2 ~ = ~ + (1—q2~/2 -F p4/4A (23)
In a plane of the variables q2 and A, for p constant;,
(22) represents a family of hyperbolas between the
asymptotes A = 0 and ~ = ~—(1 + 2cos2-yjq2 - 1/2.
We are interested in the branch with ~ > 0; (other-
wise,we would have an increase of the wave flow down-
wards, seed. Solving for A, we obtain
_ (1 +2cos27)q2 - 1 1 TSq (24)
r
Sq - 71 - (2p2 /(2 COS2 ~ + 1 )q2 _ 1 ~2 (25)
Sq iS real only for q2 ~ (1 + 2p2~/(2cos27 + 1) >
(1 + 2p2~/3; this implies that in the zones around the
stagnation points where q2 < (1 + 2p2~/3 no steady
waves can exist.
The upper sign of the root corresponds to gravity
dominated waves (A = Ag ); the capillarity dorrunaled
waves (A = En, lower sign) are better described by7
4 (1 + 2cos2~)q2 - 1 i ~ Set (26)
For any A, confluence of the two roots occurs for
A = p2/2 corresponding to the minimum of c fouled
earlier (21~. For the range (1 + 2p2~/3 < q2 < 1 + 2p2
the angle ~ is restricted through
Act< 2 arccos ( 2P _ 2) (27)
(under approach (A), fly I< arccosp, independent from
q2~. Beyond this range, for q2 > 1 + 2p2, the minimum
of Ag is no longer p2 /2 but the value corresponding to
= ~2. Then we have
O2 _ 1 1 + `/1 - (2p2/(q2 - 1))2 Q2 1 ,~_`
2
If we accept the argument that stationary waves can-
not propagate into areas where Cg/C is negative, the do-
main of admitted ~ values is further restricted (see(11~)
through
2ccg = A + 1 _ q2 + 4PA > 0 (29)
.
7The relevant analysis has been established and pro-
foundly discussed by Crapper t13] for tile no-modified
approach (A)'
160
,.% . v ~ -
P~Cnn/U~ O . ~J6 / ~ ,?
(em o,7 An to 9,,,#
~ ; "' Ash ,,~
C._ ~ 2 or (; ~ ,~ 1 .G 1. ~ t 1 4 1.(~ 1 63 q 2
~2
~= 3 Do nain D calf Admitted Waves for p= 0. 46 ( i. e. U
For gravity waves with q2 > 1 + ~p2 this implies
A > Aid _ (q2-1) (1 -F >/1- 3p4/(1 _ q2)2) /2 (30)
One may observe that this limitation is automatically
met if ~ ~ ~ 1/2 + cos2 ~ > 1 then
(see Fig.3~. Ogle inky wrote trial for any By we field from
(22),(25) and (5)
(C/U)2 = fig + ~ + (1 - q2)/2 (31)
All the above restrictions can be visualized through a
display of the dependence between the wave front an-
gle 7 and the ray direction angle cat with p and q held
constant. Mom a geometrical interpretation of (19), in-
voking the sine theorem of elementary trigonometry (see
Fig2.), we find
sin a _sin(7 ~ ~) = _s~n) (32)
and hence
tang = singly = singe `
cos ~—Uq/Cg 1 + cos 27—2C/Cg
Setting p =°,~-l, we have Cg/C = 1/2 in accord with
Kelvin's results; we find that act ~ will increase with ~~ ~
up to some maximum C:tk = arctan (1/~) and then fall
off to zero with ~ = 7r/2. We may observe that for non-
zero p, unless q2 > 1, ~ approaches zero only together
with A, as cost will remain positive. Thus an outgoing
ray act > 0) can turn inward again only if the wave
front normal changes from inward (y ~ 0) to outward
at the "caustic" (in the terminology of Yim [174) under
a maximum of the wave length due to ~ = 0.
One may consider the range of negative car in order to
leave By positive. If we exclude here those parts of curves
where ~ turns positive due to cg 1
under (A+)'),we may observe that ~ as an odd function
will in general leave opposite sign to By.
Let us refer to the range for which ~ cat ~ is increasing
from zero with ~ ~ ~ as to that of transverse waves and
define the maximum value attained for ~ in this range
as the modified Melvin angle (Xk. Then we find treat for
q2 < 1 -f 2p2 each curve for gravity clominated waves
turns unpiler horizontal tangent ~ witI-r ~ = ~r/2 q- :) into
one for capillarity dominated waves, so that after tan cat
changed sign due to increase of Cg/C > 1 for such waves,
falls oil: to zero again with the ray finally normal to
tl~e wave crest again, but opposite now to the flow (li-
rection.
Note that our
forrnulatio~s and considerations tI~rougl~out this paper
are refered to the domain around the starboard side
of the bow, where car is positive, hence By negative in
general. To deal with the other ship side, statements
remain valid if ~ and ~0, By and car are counted clockwise
against the x-axis and flow direction there. Note that,
observing 4kn/c2 = p4/~4q2 COS2 7) in (11), we have the
deviations of Cg/C form 1/2 depending on p2 and on q2
even under approach (A)', where tile term (l q2~/'2
is disregarded. Cg/C arid Thence ~ c' ~ increases wills cle-
creasing speed U (i.e. increasing p) and with decreasing
distance from the stagnation point (i.e. decreasing q)
for ~ held constant. This implies an increase of the mod-
ified Kelvin angle (which is measured against the now
lirectio~!) especially near the bow, in particular under
approach (A+)'! This is well in accord with the exper-
imental observations of Miyatat14] with wedge- shaped
models with U = 0.5m/s, (p ~ 0.462) and U = lrn/s,
(p~ 0.231) .
Away from the bow, where q2 ~ 1, Claus ~ ~ 0,
there is little difference between (A+)' and (A)'. But
again we observe an increase of Cock with decreasing U in
accord with Miyata's experiments with a rudder model
(see Inui t15~) for the speeds U = 1.15m/s, 1.72m/s and
U = 2.3m/s corresponding to p = 0.3, 0.14 and 0.1.
Irt the domain where q2 > 1 aside of the ship, a re-
duction of Elk iS predicted under (A+ )' including a ter-
mination of rays with short gravity waves with cg ap-
proaching negative values. We may retention that for a
vertical circular cylinder, q2 increases up to 4.0, whereas
for conventional forms q2 W]1 not exceed 1.2.
It is only for not too small q and for not too large p
that car is really stationary for ~ = Ark and talus marks
a transition from transverse to divergent waves, so that
with do/d: = 0 we may expect a wave cusp effect. Let
us search for the most forward point along the water
line of a ship where stationarity occurs (i.e. irnmedi-
ately near the stagnation point for pointed bows) and
trace a curve from there with dy/dx = tamed + Sky;
tibia s~ cold ~efii~e so.-,e cusp line.
~ 9oo
| q2 = 0.5 (near bow) |
capillary wage! /;wave
-90'
MU = 0.5m/s,tU = 2.5m/s (both(A+)'),- 3 - (A)'.
| q2 = 1 (far field) |
capillary wav;s
/gravity waves
· ,
-180° ~~ —90°
t
· -r
~ U = 0.5m/s,- ?-u = 1.5m/s 3 U = 2.5m/s
For q2 = 1 there is no difference between (A)' and (A+)'.
·/
cg/c negative! .2
1
q2 = 40 (aside of a thick body) l
/
,.~,.
/
/ ~ - it.
.
.
capillary waves /gravity waves
.
/
—180° 'am -90°
—(A)', .~.(A+)' (only negligible dependence on p)
Fig.4Wave front angled versus ray angle cY
161
9oo
t
or
ALPHA . v ~ C:AMMA
p=O.6
_ O
5A>1~dA
~ 2-
-
ALPHA . v 5 . GAMMA
P=0.4
it/ ,' of /
~ Z Cl/
/ ~ ;~ ~ ~ ~=~
GAt-1~1A
Fig.4 (a) a. vs. Y (for(At )')
curves over the dotted dine for capillary waves; curves
under the dotted lisle for gravity waves.
ALPHA . v 5 . GAMMA
Fig.4 ( b ) a . vs. Y
1 1
-se. oo So. oo
GAMMA
162
~ rr
ALPHA . v s . GAMMA
p=0~1
rTr
2
.~, '/ ~ 1
Hi/./' r
67Ji I"
~,.~
C~^
77
2
n
( for ( A) ')
..
. Or
;
O
The Si equation Of Ray Tracing Near A Corner
And The Short Life Of Rays Near The Waveless Zone.
Let us again consider 2-D potential flow as the basic
flow, so that complex analysis can be used.We introduce
Z—cc+iy = r·ei~y—u-iv = Uq epic, (34)
K—kit - ik2 = k e-i (35)
and
P—(9(r )2—i(9(r by =—(dV/dZ)V (36)
The P stands for some gradient of the double body flow
pressures We can write the first ray equations (12) and
(13) as
dZ 2 TV _ cs K' = 2 cat · e-i(+) (37)
With ds as differential of the arclength along the ray,this
implies that dr/ds = 1/ ~ dZ/dr I= kc/2ca`, so that we
can write ( 19) and ( 18) as
1 dK {1 dk .d~ i\ dr
K ds \
Further Considerations About The Ray Approach
-
A "ray" i:~ the sense of our analysis is defined
as a characteristic to a partial differential equ action
F(~,y,S2,Sy) = 0 for a function S(x,y), seem.
But we should ascer-
tain that the essential features of the complex 3-D flow
near the ship, including sensitive variations, can really
be modeled adequately through functions S(x,y) with
slowly varying gradient and associated complex a~npli-
tude functions A(x,y). Note that until Yim's t17] re-
cent investigations, no ejects of the Froude number on
ray curvature could be modeled, and the variations of
wave resistance only resulted from interference ejects
in tlte far field computed through integration along the
rays. Nevertheless, our in-
vestigation showed that certain global characteristics of
the wave pattern, such as the variation of Lark, and hence
of the tangential direction of the wave domain bouncl-
ary (visible in the rich stock of Miyata's experimental
results) can be predicted even near flee bow with ap-
proach (A+)'.
An evaluation of merits for the competitive ap-
proaches (A)' and (A+)' may be attempted. But
it does not seem pertinent to discrim-
inate between a "correct" and a "less consistent" ap-
proach, although it is obvious that with inclusion of sur-
face tension effects the rule of "automatic order change
through differentiation", essential for (A), can not be
maintained.
Actually, the omission of terms with 9¢,r (and hence
with 1 _ q2) under(A)' has no fundamental consequences
for our analysis in general. Certainly, the extent of zones
without steady waves ahead of a blunt bow is consider-
ably larger under (A+~; well in accord with data from
experiments with a vertical circular cylinder, for which
we have evaluated both approaches. However, the nu-
merous recent investigations on the flow ahead of a blunt
bow (see the survey by Mori t183) make clear that be-
tween the bow and the stationary capillary wave zone we
have to expect a finite domain with either a stationary
plateau, a turbulent free surface or instationary waves
propagating forward (Osawa t193), and the flow visuali-
sation experiments of Kayo et al t20] display a system
of instationary "necklace vorticies" in this domain. And
the decay of capillary waves through viscosity, as inves-
tigated by Messick and NVu t21] should be considered.
Numerical Calculations.
Our calculations have been performed both for ap-
proach (A+ )' and approach (A)'. We considered the
class of bi-circular cylinders which had already been in-
vestigated by Brandsma with conventional ray theory.
The analytical expression for the velocity potential W
is given in complex notation t12] ilk terms of bicircular
coordinates ~ and ~ through
Z = x + iy = L/2 cot(~/2) (49)
W = ~ + ilk = UL i/ncot(~/n) (50)
with ~ = (+ir~, ~ = b~ -52, rig = In(
Thc lays of gravity- find c.tpill`~rity- waves in the civics
of different body advance speeds and entrance angles arc
shown in F-ig.6, 8, 9, and Fig.10
To show the difference Hewn ours and the conventional
ray Amy and ~ pan of char Run, fcwar~l- (dcw~nun)
and backward- ( upstream from the far fickl ) tracing based
can convcnticx~l ray theory are also pcrfonnod. For the latter,
considering the uncertainty due to the stagnation point, rays
arc backtr.tcod from the far fickl towards the bow, iterately,
by changing initial conditions, the results in the far field
by usual ( forward tracing ) method being used as the first
set of which, till ray reaches a given- sized ~,cighborho~xl
of bow stagnation point, and wave angle converges at the
same time. the results are shown in Fig. 12.
In all the figures of lays prcsentod, the pairs of members
attached to each ray give ~ ,. arid ~ ~ the initial arid filial
values of ~ ( in degree ) . Thc short segments on the rays
show the local wave fronts.
To get better accurracy, step widths in ray tracing with
Runge- Kutta method were carefully chosen, and the reliability
was checked by halving the step widths.
Conclusions
Ill the course of an assessement of the ray approach
with regard to representing essential features of the wave
pattern, we have incorporated capillarity ejects in our
analysis to overcome obvious shortcomings near the hare
stagnation point.
~~_..~,.e," ,&~A ~,`~ =w w
Our formulation AL+ )' (eq. (2) ~ generalizes approach
All', where certain terms related to the doubl~-tody
flow pressure are disregarded. Although within our
work we could neither provide a rational model for ray
generation nor even a justification for extending the ray
approach the hull surface vicinity, the following facts
have been discovered or confirmed.
Our numerical investigations have displayed several
global effects on the wave pattern geometry result-
ing from the inclusion of capillartity to our analytical
model; they gain practical relevance for small speeds,
say for U=2 m/s (if we consider a minimum capillary
wave speed of 0.23 m/s.~:
(1) Both the far-field Kelvin angle and the "modi-
fied Kelvin angle" near the bow (i.e. the angle between
tangents to the wave region boundary and to the hull
water line) are found to increase with decreasing U.
(2) With increasing bow entrance angle, both the zones
of no steady waves and the surrounding short-life belt,
from which no rays proceeding into the far field can be
found, grow in size.
(3) The outward extent of this belt is decreasing with
increasing U. i.e. the stronger the capillari.Ly
the lar~,er the snort life belt.
The above findings are in qualitative accord with
some tendencies one may observe from experimental vi-
sualizations of flow and wave pattern as presented by
Inuit15] (his Fig.2-2 is reproduced in our Fig.7), of Miy-
atat14], of Maruot7] and of Osawat19; (see Fig.11~. It
is true that we can not expect our analytical model to
cover all features of the complex phenomena observced,
ejects of viscosity and finite wave elevation in particu-
lar, though the latter may be assumed to be less signif-
icant considering the low speeds of the models. Thus it
seems that in this regard ray theory displays a certain
value for predicting ship wave phenomena, although the
re-entrance of rays or their reflection at the water line
must be considered an open problem, among others. In
any case, the authors would like to emphasize the need
to take account of surface tension at low speeds, well in
accord with Maruot74. We hope that our work reported
here can add some further weight on this aspect
Acknowledgements
The authors express appreciatiol! and gratitude to
the 13eutsche Forscb ungs- und Versuchsanstalt fur Luft-
und Raurnfahrt for sponsoring the first author's one
year research fellowship at IfS Hamburg. Our thanks go
to faculty and staff of IfS for all their kind assistance.
The authors would like to dedicate this paper to
Prof. T.Y. Wu to the occasion of his sixty fifth
birthday.
165
References
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flow."J. Fluid Mech.Vol. 9 pp 337-364 (1960)
2 Inui T. and Kajitani,H.:"A study on local non-linear
free surface effects in ship waves and wave
resistance." Schiffstechnik Vol.24 pp.l78-213 (1979)
3 Keller,J.B.:"The ray theory of ship waves and the
class of strea~rdined ships."J.Fluid Mech.
Vol.91 pp.465-487 (1979)
4 Yim,B." A ray theory for non-linear ship waves and
wave resistance."Proc.Third Intern. Conf. on Num.
Ship Hydrodynamics Paris pp.55-70 (1981)
5 Brandsma,F.J.:"Low Cloudy number expansions for
the wave pattern and the wave resistance of general
ship forms." Thesis T.U. Delft lllpp.~1987)
6 Eggers,K.:"On stationary waves superposed to
the flow around a body in a uniform stream."
IUTAM Sympos. on nonlinear water waves, Tokyo.
edt.by K.Horikawa and H.Maruo.
Springer Verlag pp313-323 (1987)
7 Maruo,H.: and Ikehata,M.:"Some discussion on the
free surface flow around the bow." Proc. 16th. Symp.
On Naval Hydrodynamics Berkeley pp 65-77 (1986)
8 Eggers,K.:"Non-Kelvin dispersive waves around non-
slender ships."Schiffstechnik Vol.28 pp.223-252 (1981)
9 van Gemert,P.H.:"A linearized surface condition on
low speed hydrodynamics."Delft T.U. 26pp (1988)
10 Eggers,K:"A comment on free surface conditions for
slow ship theory and ray tracing."
Schiffstechnik32 pp42-47~1985)
11 Longuet-Higgins M.S. and Stewart R.W.:"Changes
in the form of short gravity waves on long waves and
tidal currents"J. Fluid Mech.8 pp566-588 (1960)
12 Milne Thomson:"Theoretical Hydrodynamics"MacMillan
13 Crapper,G.D.:"Surface waves generated by a travelling
pressure point."Proc.Roy.Soc. A 282 pp.547 558 (1964)
14 Miyata,H.:"Characteristics of nonlinear waves in the
near field of ships and their effect on resistance."
Proc. 13th Symp. on Naval Hydrodynamics
Tokyo pp.335-353 (1980)
15 Inui.T.:"Fiom bulbous bow to free surface shock waves-
liends of 20 year's research at the Tokyo University."
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Journ.Ship Res, 25 147-180 (1981)
16 lulin,M.P."Surface waves from the ray point of view."
The Seventh Georg Weinblum Memorial Lecture.
Proc.14th. Symp.on Naval Hydrodynamics
Hamburg pp.9-29 (1984)
17 Yim,B.:"Some recent developments in nonlinear ship
wave theory."Proc.Int.Symp. on ship resistance and
power performance, Shanghai. pp82-88 (1989)
18 Mori,K.":Necklace vortex and bow wave around
blunt bodies." Proc.15th Symp. on Naval Hydro-
dynamics Hamburg pp303-317 (1985)
19 Osawa,K .: "Aufmessung des Geschwindigkeitsfeldes a
und unter der freien Wasseroberflache in der Bug-
umstromung eines stumpfen Korpers Bericht Nr.476
Institut fur Schiffbau der Universitat
Hamburg,125pp(1987)
20 Kayo,Y.,Takekuma,K.,Eggers,K.and Sharma S.D.:
"Observation of free surface shear flow and its
relation to bow wave-breaking on full forms."-
Bericht Nr.420 Institut fur Schiffbau der Universitat
Hamburg,33pp(1982)
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surface waves generated by steady disturbances.
Rep. 85-8 Engg. Div. Cal.Inst.Techn. 31pp(1958)
166
9p, idAeF ENrRA^~£ ~E A. .~2. 5
o
o
(a)
8
S ~;' 0',60' 01,9~1.1''l'001 ,1,20'
l ~ ~ ~_
~ 0~
( b )
-0
o
o
_ ~F ENrFA~= ~' 4. -22. 5
l
-
1
_
1
~1
1
1
1~- (d'
t
o
G
Fig.6 Rays of gr~lvity W`IVCS for cl bicircLIlar cylin~icr
with half a~trance angle Ae= 22. 5°, in cases p= 0. 6, O. 4
and 0. 1, corresponding to U= 0. 39, O. 58 and 2. 31m/s
respectively, for Gn= 0. 231m/s.
(a), (b), and (c) are based upon (iL+)';
(d) is ba~ed upon (A)'.
~ tracing i ~ s topped i f the ray ent ers
the bo dy ~
· 60 2 c
~-W \ .
~ '~_ ~ ~—a~ ~ ~
p.O.1
Fig.7 Wave pattern of a rudder model of length ().3II
with U = 0.65' 0.5 and 0.34 m/s (by courtesy of Prof.
T.1nUi from [15])
(Iilor this rudcler, Pn30.4' p=Oe33;
I?n=0 e3 ~ P=(). 45;
F.n-0.2' p=0~66)
167
pa D.37
6 0
0 In
~ p=o.25
rat
to'
,,_--~~~~~--_,~
o
-
o
p= o. 37
jp=o,25
Fig.8 Rays of bow capillary waves for different cntrancc
angles and body speeds. ( based upon ( A+ ) ' )
~ U= 0.62 and 0.92 mj~s )
D
o
o
o
c,
168
/ ~
16.],.= 3,
te. 1. ~ ~ ·/
~2~ ~ D1
_: ;F fi~
r ~N-~
r 3: ~. .J J ^
~ ·~4~2~ ;~~~;;\> ~ ~ ~ =
r~ '~ C. sa f. -
o
-
o
p(c-/U1~0.~00 ~ = `) 4 ~
(l b 29 5 · · ~
]= ~
PtC-'Ul'0.100 ~ = 0 ~
Fig.9 Rays of gravity waves for diffcrcnt entrance angles
and bc)dy speeds. ( based upon ( A+ ) ' )
~ l,r= 0.58 and 2.31 mys)
~::?
C j a:::`
Fig.10 Rays of bow c~piLary awes in Wont of a cir-
c~ar cylinder (moving to the hO) under approach (~'
U = ~ ~ ~ ~
Fig.l] Sow waves ~ front of ~ lacing
cylinder with dinette D= 0.~ m' moving to
the left' at the speeds of U=O.61 Oe7 ~ 0~9
m/s (From Ee OBEY L191
169
o
°_ HALF ENrRA~E A~f A, - 22.S
o
o~
srAQr ING P r . OF RAYS. I XO. YO I <0.0000)0,0050)
(a)
o
O l
20 0 ~ ~ ~ -i: .sl
O o · 5 2; ~ - ~ ~ / _~
A
_ ~ \ -es.:
- o 00 0. 20 0. ~0 0.60 0.60 ~ . 00 1 .20 ~ . ao ~ .60 ~ . SO 2. 00
X
o
° - HALf ~NTRA~E A~ Ac-22.=
~ACKV - OS-TRACI~ ~ RAYS
_ 67 ~~
, -8t .7
~ ,_- O
66 t8 6'27 23 l - - ,' _ ~ '/
g
o
t0_
o
>~ -
o
_
o
o
. _
o .
1
_t
01
o
-o
~' ' ' ' 0 60 ' 0 80 ' 00 '
'.20 1.40 \.60 '.eo 2.00
'o' 26 ~-~2is . ~ P=0 .4 ~
--;~ I -1 1
~_=: -------- X 1 . 23 1 ~ ~ 60
_° ~_ ~
i I I I I I
- 1~.80
_` \ ~ ~
209'S7~2;5~9 272 3 //~
4 s; 34 '3 ¢.////
3 ' .~ - t ~ //
1C a ;~7 O -'
Fig.12 Rays traced with ap-
proaches based upon conven-
tional ray theory:
(a) Forward~downstream~tra
cing;
(b) Backward (upstream)
tracing from far field.
Rays ~hat enter the body
have been removed )
Fig.13 outer boundary of rayn
of gravity waves, traced
with dy/dx=tan (~+: ), based
upon ~A+; .
Il.B. it might not be the at-
t ainable boundary of rays
of gravi ty waves; and, it
is not a boundary between
ray families of gravity-
and capi llary- waves,
p=O.1 ~
170
o
. ~
r n ~ ~ I 1 nl,,il ~ 1,nl ~ l
/
GoO,~,2O,:
1C.O. 55 7 ~
t t . · -57 . 9 '/ o
1
o
-
o
o
8
.~
og
o
og
24
~ 2.'i=.'~'-i~
~2~ '''''"--
. ~0.2O 0.40 0.60 o.6~L-l" lToo ,1 Ir20 ~ 1r40 ~ 1r60 ~ r 1 -
-a ~~
PtC.~]'0 1nn
_~' ''-a.
, . . . . ~ . . . . . . .
O ~ I I · , , . , , , ,.-, . . .
O ~? O.bO 0.8~-' 1.00 1.20
~ is shown negl gable.
.~. - ..
_O .0 '. ..; O' a/
Pt c. But 0. 100
Fig,15 Rays of bow capillary waves
(based upon (A)' ~
Fig.16 Rays of gravity waves
(based upon (A)' ~
. . .0 · .60
Fig.17 Precision check for the tracing
technique, the upper curves are obta-
, , , ined by halving the step width used
1 . SO 2 CC
for the bellow curves, the difference
171
DISCUSSION
by H. Kajitani
1) I suppose the ray tracing is a kind of
low speed theory. I'm not sure that a pretty
high Fn applied in Fig.7 is available or not.
2) Could you comment on what difference
can be observed on the traced characteristic
lines between with and without surface tension
effects?
3) The wave length of capillary waves in
front of a ship bow changes with its distance.
Have you computed the capillary wave phase?
Author's Reply
Prof. Kajitani's worrying about applying
ray theory for high Fn is certainly natural.
We use Fig_ 7, the highest Fn is 0.4
there, (from Inui and Miyata) to show the
qualitative confirmation with the test
results We don't think that ray theories
(at present) can predict strong non linear
effect. Keller[3] claimed that ray theory may
be useful for Fn <0.7. We are more conserva-
tive in this regard. As to the differences
between those with and without surface
tension, they could be listed in Table Al.
We have not yet calculated the capillary
wave phase. It could be carried out through
integration.
DISCUSSION
by H.S. Choi
First of all, I would like to congratulate
the authors that the surface-tension effect
Table Al
Conventional ray theory
(no surface tension)
1 Point disturbance, all rays are
from stagnation point; Stationry
waves exist even near the stag-
nation point.
2 Ray and wave patterns are inde-
pendent of body steed U.
has been successfully included in the ray
theory to clarify the wave pattern around the
bow more clearly. It may be more usefully
applied to a small-scale models. If it is the
case it is possible that the local phase
velocity reaches to the minimum celerity of
capillary gravity waves (=25 cm/see) and the
capillary wave breaks. It implies that a new
source of singularity has been invited to your
method. I would be happy if you comment on it.
Author's Reply
Thank you for your comment. If the wave
length of a gravity wave is decreasing when
progressing, the wave might break before the
local phase speed c reaches the local minimum
phase speed cm. In our approach, we start rays
of gravity waves from the short-life-belt
boundary, where exist the shortest omitted
gravity waves. The waves seem to become longer
when propagating (cf.(48), which shows that
near the bow, 1/k dk/ds < 0, that means
increasing wave length along rays ). If c
makes the mode for the ray pattern more
accurate than the one described by Brandsma
and myself. It is a pity that the authors do
not say one word on the influence of the wave
excitation coefficients and the corresponding
wave amplitude. It is my philosophy that one
must try to balance all components of the
building. To my opinion one approach has such
a balance at its own level. Do the authors
expect that our approach to the amplitude
problem is applicable in this case? If so, do
they expect that the influence of surface
tension is noticeable there just as well.
Author's Reply
Thank you for your congratulations and
comments. The aim of this paper is to find out
if surface tension is taken into con-
sideration, the ambiguity and difficulties of
the conventional Ray theory, as found by many
others, can be overcome. We would not blame an
existing theory for not being perfect. On the
contrary, we appreciate every effort made by
previous authors who developed ray theory and
made it possible to apply it in practice. In
view of that some important features of the
real world can not be predicted with
conventional ray theory, we think that some
improvements may be necessary.
We have not yet calculated amplitude. Our
concern in this paper is on ray pattern. Our
results show that surface tension may not be
disregarded for slow ship problems, at least
in small scale cases. Even if it turns out to
have no significant effect on the final
results in some cases, it can still be used as
a way to circumvent the difficulties in ray
theory. From the viewpoint of validation, the
assumption of infinitesimal wave length at the
stagnation point is always an unpleasant
thing. We tried to get rid of it.
173
