ADA169794 - Second International Conference On Numerical Ship Hydrodynamics - September 1977 - University of California, Berkeley
ADA169794 - Second International Conference On Numerical Ship Hydrodynamics - September 1977 - University of California, Berkeley
ADA169794 - Second International Conference On Numerical Ship Hydrodynamics - September 1977 - University of California, Berkeley
C.D
c
ic
ELECTE
JUL 15 IM
IN,
cw s-rATEmENT
Appm-od bw pubhc rel"
Dwbution uniin3itw
86 7
065
The
t'roceediflgs
International
Conference on
Numerical
Ship Hydrodynamics
Edited by
John V. Wehlausen
and
N i Sat ven
Sponsored by
The" David W. Taylor
Center
Naval Ship Research anidDevelopmnft
The Department of Nevat Architecture
Cottege ot Engineering, and Univertity Extension
University of California, B~erkeey
The Office of Naval Research
September 19-2t,lgfl
SEL-.EGTE0
iln
nOGER BRARD
DEDICATION
These Proceedings are dedicated to the
memory of Roger Brard. He expected to attend
this Conference together with his wife Th6r~se
and was to take part In it both as a speaker and as
a chairman. A short illness ended with his death
on July 15th, a culmination of many years of
uncertain health.
Roger Brard's involvement with ships was both
broad and deep, touching upon almost every
aspect of naval architecture. However, shiphydrodynamic problems were an ever recurring
theme, and in recent years he devoted his efforts
principally to these. Earlier he was also known for
his contributions to probability theory and he
regularly lectured on this subject during his years
at the Ecole Pirlytechnique. Few of those who
know only h!:' recant work on resistance and
maneuverability or his earlier pioneering work on
pitching and rolling of ships under way are likely
to be aware of the breadth of his activities in ship
theory, fluid dynamics and mathematics. In the
following paragraph we try to give some idea of
thisbrsadth.
Brard's doctoral thesis (In rrathematIrs) was
"On some oropertles of the geometry of ship
hulls" (1929).The following year there was a paper
on the rohatilg of boiler feed water and the year
after that. one on the exhaust of internal
combustion engines. To give a further Idea of the
almost Incredible extent of his activities, a
sampling of topics on which there are published
papers follows: theory and design of propellers,
acdd mass in rolling, effect of added mass and
wave generation on stopping of ships, nonlinear
oscillations, statistical theory of turbulence, the
law of large numnbers, siationar) random
processes, model testing of towed barges,
representation of hulls by source-sink distributions, -ell-propelled model tests, sea trials of varl
ous ships and comparison with model tests, flow
about deformable profiles, cavitation. And of course
the topics mentioned earlier The list Is not exnaustive. Furthermora, during 1932-38 while at
the Arsenal at Brat lre was In charge of repairs
w ieveral cruisers, of the construction of another.
arid of the arniament of three baltleships.
His talents and contributions were not
unrecognired in France and abroad We mention
first a few of his prizes, chosen to show their
diversity, a medal from the Office National des
ORDERING INFORMATION
Sec
Firt ilererinCofrenc
I
oI Cual
ShpHdoa nmc
01*
___________-
Acalson For
NTIS GRANT
TA
ItiL
ntori.
Jitt
~-
Distributionm/
International Standard Bookt
Number: O9lM17.
University Extaeson Publications
UnIversitlyofCalifornia
Berkeley, CIllfornl9472O
IVI
D181;
SPOO it
PREFACE
Although theoretical ship hydrodynamics has made steady if not dramatic progress during
the last thirty years, it suems evident that this progress has been achieved for the most
part by application of various methods for ieplaclng the difficult nonlinear equations by
approximate ones, In particular by the use of perturbati,n methods. The simultaneous
spectacular growth of computing machines and ,he;r impact on the development of
numerical analysis were not ignored, but their influenre on ship hydrodynamics was
primarily in widening the possibilities of computing derived formulas and almost not at all
in altering the methods of approach to the problems themselves. It had been evident for
several years that the time was appropriate for taking formal notice of this fact and of
encouraging wider participation in ship hydrodynamics of persons with stroig beck.
grounds in numerical analysis and computing. The First International Conference on
Numerical Ship Hydrodynamics was In risponse to this need. it brought together researchers
of rather d;verse backgrounds, with the aim of giving to each a forum for his own approach
to ship-hydrodynamic calculations and of providing opportunitie for interaction.
This first Conference, held in October 1975In the National Bureau of Standards InGaithersburg,
Md.,was by all criteria a notable success. The papers were interesting and the opportunities
fo; discussion, both formal and informal, were adequate. Furthermore, the Proceedings
appeared in a remarkabiy short time as such things go. It was evident tMlata second Conference was both deslrablo and desired. The present Proceedings are the result of this
second Conference, held in b,.' -ley on September 19-21,1977 and sponsored jointly by
the David Taylor Naval Ship Researc, 1nd Development Center, the Office of Naval Research
and the University of California Berkeley. .'en mere mention of the O'fice of Naval Resech as
a sponsor does not do justice to the condlntius, effective and informed support that it has
given to the development of numerical methods in ship hydrodynamics in recent years.
Without this, neiltier Conference would have taken place, and ntch of the reported
research would not have been done. The .ctual organization of the meeting, aside from
selection of papers, was -arried out by the Extension Division of the University through the
capab'e hands of Linda Held. Since authors provided manuscripts In final form for reproduction, only miner editorial changes were possible aid no attempt was made to achieve
consistency in format or reference style.
Finally, it seems appropriate to call attention to the remarkable diversity of the papers.
Even when the same problem is being treated, the methods are different. It was one of the
purposes of the Conference to encourage this diversity and to bring the results into juxtaposition. Diversity extended beyond subject matter. Over a dozen countries were represented
among the participants. Such Juxtaposition Is Also important.
DedIcatlon ..................G.0N.T
S.....................
Preface .............................................
KEYNOTE ADDRESS
Numerical Ship Hydrodynamics, Then and Now. L. Larn. ..- br, University of Iowa ... .......... 1
SESSION I: THREE-DIMENSIONAL SHIP-WAVE PROBLEMS;
Chairman: J.C. Dern, Bassin d'Essals des Carhnes, Parie France
Invited Paper: Survey of Numerical Solutions for Ship Free-Surface, Problems, R.B. Chapman .
Numerical Evaluation of a Wave-Reafatance Theory for Slow Ships. E. Babe and M. Hera.
5
17
...........................
57
K.J Sal .. . . . .
78
- .. .. L'
- .. . .. . .. .. .. ..
. . . . . . ..
. .. .. .
.. ..
88
04
PART11
...
107
...
124
136
57
Integral Equation Methods for Calculating the Virtual Mesa InWater of Finite Depth.
..
P.Sayer and F.Ursli..... .........
.....
...
Thelory of Compliant Planing Surfaces, L J Doctors
Diasussion Part 1,Soosio,, III Papers: HrMarve, University of Yokohama, Japan, Invited
. . ..
. . . . . ... . . . ..
Discusser
vi
100
17
185
tI-S
PART 11
Numerical Solution of tha Navter-Stokes Equat ion for 2.0 Hydrofoils In or Below a Free
Surlace, SI'. Shanks and J.F. Thompson ....... ....................................
202
Finitb-Dfferenco Computations Using Boundary-Fitted Coordinate Systems for Firest-Surface
Potential Flows Generated bySubmerged Bodis, H.J. Haussling and R.M. Coleman.....-221
Discussion: Part 11,Session III Papers: J.W. Schot, David W. Taylor Naval Ship Research and
Development Center, Bethesda, Maryland, USA....... ...... ......... ................
234
............
258
An Application of the Boundary Integral Equation Method to Cavity and Jet Flows, B.E.Larock 2tr9
Discussion. Session IV Papers: M.P. Tulin, Hydronautics, Incorporated, Laurel, Maryland,
USA, Invlted Discusser
275
1278
...
......................
285
292
...... 301
PART 1
Finite Element and Fiift Difference Solutions of Nonlinear Free-urface Wae Problems,
GM. Yen, K.0. Lee, and T.-~ Akel.................
........
305
Transient FreeSurface Hydrodynamics, M.J. Fritts and J.P. Boris .......
-,..31g
Discussion. Pont1,Session V Papers: A.J. Hernians, technische flogeschool Delft, The
Netherlands, Invited Discusser ..
-....
..
.. ... ....
. .-
329
332
341
BR. Penumalll.
..
358
i...l..l7
396
List of Participants
.....
vii
397
Schoenhe-r
made
lmjsrtant
pealed
consultants,
Alex Wie-
uate utudy.
greatly
of ship hydrodynamics
war y.ars.
of
liediete
seeking
practical
to the
it the post-
to the year.
sols! uin
to
Basin,
l.i
Nor was the nmomentumof University contri1iutionsto ship hydrodynamics lost after the
war, through the activity of a new organization,
The Office of Naval Research. By sponsoring
research at other institutions, participating
in the organizing of symposia, such as the
present one, contributing to the publication of
majir works and journals such as the Collected
Papers of Havelock and the Applied Mechanics
Reviews, and establishing international relationships b,tween researchers, the Office of
Noval Research has been an important factor in
promoting interest and productivity in ship '
hydrodynamics. Subsequently. the Taylor Modc
Basin, under its GHR program, also undertook
the spcnsorship of ship-hydrodynamic research.
Before the War, there were practically
no outlets for publication of research in
applied nathematics or ship hydrodynamics. The
Transactions of the Society of Naval Architects
could hardly serve this purpose since it accepts
few papers each year, and these were, rareLy
This has been
of A ship-hydrodynavcc nature.
called the "integral gap'" in the Transactionls.
The appearance of several new journals, such
as the Qu.rterly of Applied Mathematics, the
Joural of Rational K-chanics, the Journal of
Ship Research, and the Journal of ilydronautics,
Indeed,
has eased the luslication problem.
with the recen announcement of several new
'uii iccatLons , there is dangler thst w may be
oveihemisd by too much literature, and overburdened referres may lower their reviewinq
standards.
orgsniration was
at
while they pursu1uedtheir graduate stud.'s
In my opinion, this was a wise and
Berkeley.
far-sighted
policy which
Coputerlies
havtincrer-
an e.
Sic other ppllers also em4)loy fiitsdifferance mthoda, but for [the Lrrotatbocal
including free-surface effects and vortex formation, and the wavewaking of the ship in the
presence of the boun-ary layer a d wake, the
problem will continue to confront us.
Abstract
Recent methods fornumerical solution of
free-surfic(! problem in ship hydrodynamics are
reviewed. ')ubasi-analytic
methods which model
a linearized free surface outside a body of
irbitrary ,hape &,e discussed in detail for
body motion problem. Also reviewed are methoJs
for the wave-making of translatig threedimensional bodies an. two-dimensional methods
for nonlinear 'low.
In..,roduction
Interest in numerical methods for solving
Free-srfAce problems in ship hydrodynamics has
grown rapidly. These methods can eliminate
ma.: of che approyimations necessary for anflytic solution. Capabilities such as a-bitrary
h,-ll
form obviously extend the practical value
of any method. The most useful ert..ods
may not
be the most general ones or those with the
fewest a.,tions. however. More limited
techniques such as the quasi-analytic methods
discussed in thispaper can be better design
tools for large numters of problems, while more
general methods can check the validity of approxipations. The purpose of this paper is to
describe briefly some recent advances in naT.icalship hydrodynamics and to suggest likely
areas for the ner future. Three general areas
which will be reviewed Are:(1) quasi-analytic
methods for body motion, (2) numerical simulat
tionof the wave-making of translating hreedimensional ship hulls; and (3)two-dimensional
metfods for highly nonlinear flows.
Lxasi-Anytic Methods
The gap betxtvn numerical and analytic freesurface metho% Il. ship hydrodynamics is bridged
by a class of methods which com-ine analytic
linearthed free-surface solutions with namer'(ally
genieral
representitions of the body.
wuasi-analytic
met'lods 111 be defined b. these
twochar#-teristics:
1))Analytic linesrized free-surface
need
eliminate
representations
4 largetheYolum
the flou over
solve
suroeundi
the obody.
(2) The body boiundtry
i, satisfied exactly
o, an arbitrary hull.
tr
+t
mL
he lower half-
iZEgq(6)
+ji(xyti
s (x'y,5) Of(.)C) .%*8(X.Yt)
SSF.
6,I-
WI
----
_T
40.4.1
Rw(-s.1j0B
%(XY~t)
Bi~x.y't)
FIGURE1. GOE
nxy
peiid
owe
SIT
Th
l
= 0
(v,y) c SFe
(7)
(n,y) c SFi
(8)
oxthews
u~~~~~~~~~n
~xyt
((3))
f
ex~~)e~~yt
(9)
where h(v,t) ropresents the fi~e-surface infisence of the hodypotential,
xt)
hYyt
(1)
vn vB*n
I(v
)v,0)"
)Q
(o,0)v SFe
B(~~)=
xy
~
~
nyt)
ed
xy
nd
(u"y
(X,..t)
nbody
0
X
(4).
(10
v g ate
at
1cvanishes
Sln
(11)
od,
For the body source and the hod"dipole ne,
tf~n~y,t) is represented by the Integral of1 the
(2)
x cR
(10)
of a time-dependent disturbance
cal prollees
acting on a linearized free surface and can be
solved 6nalytically in terms of h~n,t). If
0)xyt en Istiedrvieaeiiily
tero for example,
B~~~)
- 0
~i
(xo,t)
V20)o.y.t)
(5)
-t
.y. l
y.I
..
4(
()
donend
while
G(X,X1 )
--G(X .y.F )
(14)
i)
(Y-0
(15)
Assume for rnowthatthe fiuid is unbounded.
Then from Green's theorem the body potential
~(")(16)
G(x,x 1) dS
)
k(Y+Y
1
j-i-,I
(17)
Thisis equivalent to a source distribution
over SB and a Cipole distribution over t 8 . The
body boundary condition for bj is arbitrary.
The most conon assumption is thatthe potential
is cntiuou
he odyThus,
acrss
B
18)
In thiscase, the norml velocity is discontinanus across the body surface and the body
component of the potential is of the form,
fof
03(x~t)
JO(-x
1,t) G(x-lx1 ) dS
(19)
SB
which Is equivalent to a source distribution of
strength 0(nxt) over the body surface,
Body _ource Method
The body source method was applied by Frank
[31to theproblem of harmonic oscillation _f a
two-dimensional shipsection. In thiscase,
the body boundary condition (13)may be represented as
12.11o
*0iiE*
(.t) 61"Al.
Uh *(i
12
/aI(-*
ds
as
T%
1 (ipt)] sl2T9
Be(' t) 00
0i(Xt)
1.1
(23)
G2xB
(i~
(22)
SB
SP
xFQ
OBi(
x t)
~i
j
58
011xil=
=
g_ DB (j)21
*s
20
S,(20
(yv1 )
k
satisfies
9.
a -L 0(,
i
;,
-9 -L y~i.;)
(25)
g1
-X
(27)
()
x ES
(28)
G 4IF -r1
r
x x ) dS (30)
3nI G(xx
SJD(xlt) -B
where D(x) is th2 stiength of the surface dipole
distribution. This body potential *B(i,t) is
ss
body
distiuous acrs
continuous noral velocity. As in the source
OB(it)
(29)
where
is the distance betweenjoints x andt
i
ofn
acrssthe utndeetedwree
srache. Te
of x, across the
~~
Vn(,t)
x f Ba .
*i(x~t)
n 11(t).
(32)
1ci,!,t) as
I
(33)
Bodi_Dipole Method
1 (xt)+
IG(x' 1 )
JD(i,t) ---n1
(34)
dS
dary condition is
()
x
).
(37)
1
o
x)
(38)
and
D
5
(39)
x C SW
ix)
E(0)
C8, fq ( )
ix)
11
(40)
(x)
111
(41)
kmO
and
NL
kEt k
-(W
' - -'(,,-'-
as1(
aty))
sa u ss
M_
1 as.
( )
k (x)
(()
iwt
-i
= Re{c(x) e
p(x,t0
Hybrid Method
methods,
For the body source and body dipole
the body
singularities are distributed over
surface, with the free-surface potential represented by the free-surface portion of the
Green's function 0hich is also integrated over
the body surface. Another method for linear
harmonic oscillations of a body immersed in a
fluid is the hybrid method described by Bat and
Yeung [9], which matches a numerical solution
in a small region surrounding the body with
analytic solutions outside this region. Bai and
Yeung [9] give two alternate techniques for
implementing the method; a finite element varlational method and a surface integral method.
The surface integral method will be briefly
described in the form developed and applied by
Yeung [101.
G(x.y.,.n) - I log[(2-)
i36)
F S
Vn(()
Jsas
-4us ,(;,,
It
[:lf(i,)
)
Oi2.2
1
as
(42)
k Xi1
-n
can he eliminated,
. 23s
;i(.0
~
30 (;.,-.
;J)
(,i-
wt
S,
t',
a.___
__
written as
SE
SL
1 (x-t) = Vn(x,t)
S Ial
b(,x) V it)
b,
(44)
j=l
IESB, t>O
;n
t)-- o
x E SW ,t , 0
(45)
)2
)-j (X,fi
g --' (i,t)
S FF
0
SF. t '. 0
(46)
I.xt)are both
(xt) - g-
SW ss
d3 #S
,(.t) ( '
( i') "
, t( .
, .-I
8US
Sw
(41)
10
I1 2
on = ( k) 1
is the frequency. This finite
summation is, in fact, an approximation to an
infinite integral with an error which vanishes
as the wave spacing tkn mTknvl - kn approaches
zero and the maximum wave number kn , which
corresponds to the shortest wave length,
approaches infinity.
(48)
48)
Vj(t)k
=l
'
t)
n(xt) = -
1at
(x,O,t)
(52)
--
F(i,t)
tt
F(i-N
NW n
iknx
= n-- A (t) e n
(49)
N+NF
where An(t) is a complex vector. The free surface is thus represented by a pair of complex
vectors, An(t) and Bn(t). The linearized free
surface equation (9) and the dynamic condition
(52) can then be expressed by N pairs of
simple differential equations of the form
3~My
di An(t)
k", -NF.u
(50
dc.,}
ynlt
kn ( '
vy
n(t)
Cn(t)
(54)
an An(t)
(55)
An't).
0...
NW
iknx
h(x,t) = g F_ Cn(t) e n
n=l
((50)
(56)
These equations can be integrated analytically for a series of small time steps over
which the body source distribution, and therefore
C2 y(t),
are assumed to be constant. Since
the bo
potential
is expressed numerically as
where Ct)
represents the body source potential
in the ?orm of the harmonics of h(s,t( defined
in (9),
-gNw
Bn(t)
(53)
(51)
(One
where n:(t) It the complex amp'litude of the nharu, 1c. k,. Is the wave numh.r. and
i_
resistance induced by the surface source distribution calculated for the flow past a Series-60
hull in the zero Froude number (double hull)
limit by Hess and Smith [17]. Although this
method neglects the effect of the free-surface
disturbance on the body boundary condition and
therefore on the source distribution, it should
give a strong indication of the effect of satisfying the exact body boundary condition for low
Froude numbers. The experimental residuary
resistance was compared with predictions from
thin ship theory and the surface soirce distribution over a range ot Froude numbers. No
improvement was obtained by placing the sources
on the actual surface. Breslin and Eng point
out that part of the disagreement between theory
and experiment may be due to interaction between
viscous and wave-making components of resistance.
Wave-Making Problems
The problem of a body translating with steady
speed through an inviscid fluid with a freesurface is usually solved for the purpose of
estimating the resistance due to wave-making.
Also of interest are sinkage and trim forces
and the streamlines over the hull, including
the wave profile. If the body is well submerged, the linear free surface is valid and
quasi-analytic methods are applicable. With
modification, basic quasi-analytic techniques
discussed for body motion problems can be
applied to a translating submerged body.
The dipole method was applied to a surfacepiercing ellipse by Chang and Pien [6] with a
formulation equivalent to a submerged body with
a flat too an infinitesimal distance below a
linearizad free surface. This formulation
contained the two-dimensional equivalent of the
line integral for three-dimenslonal surfacepiercing buoies.
12
low frequencies (Froude numbers), to the nonlinear portion of the damping coefficient--an
effect closely related to wave-making.
The limited results row available indicate
improved wave-making resistance predictions by
perturbing about the double hull flow, even at
moderate Froude numbers. This suggests that
horizontal transport terms in the free-surface
conditions are important factors in the wavemaking resistance of realistic ship hulls.
It should be expected however, that in the
moderate to high range of Froude numbers the
horizontal velocities on the free surface will
cease to be modeled by the double hull flow.
Two-Dirronsional Nonlinear
Free-Surface Problems
Fundamental aspects of nonlinear free
surface flow are most easily studied in
dimensions. A few nonlinear solutions of
interest for ship hydrodynamics will be mentioned in this section. One class of problems
is steady nonlinear flow past a two-dimensional
disturbance. Problems of this type, including
flows past submerged vortices, flows past foils,
and shallow water effects have been studied
extensively by Salvesen and eon Kerczek (35].
[36], [37]. Of particular interest are
comparisons with perturbatior theory and with
experiments with a foil. Nonlinear effects
contained in seconu and third order perturbation
13
expansions are evident !n the numerical nonlinear free-surface elevations for flow past a
submerged vortex. For positive circulation,
third order theory provides an excellent approximation to the nonlinear wave-making resistance.
The basic method is to repeatedly solve Lapiace's
equation and correct the free surface elevations
for the error in the dynamic condition until the
solution converges. In this mannt, Salvesen
and von Kerczek could generate waves near the
maximum experimental steepness. Other itivestigators have generated steady nonlinear flows
with unsteady methnds using small time steps.
Haussling and Van Eseitine [12], [38], for
example, have applied both spectral and finite
difference methods to steady flow past a pressure patch.
14
References
1. Bai,K.J.,"A Localized Finite-Element
Method for Steady, Iwo-Dimensioial FreeSurface Flow Problems," First International Conference on Numerical Ship
Hydrodynamics, Gctober 1975.
1963
21. Wehausen, J.V.,"Use of Lagrangian Coordinates for Ship Wave Resistance (First- and
Second-Order Thin Ship Theory)," Journal
of Ship Research, Vol.13,No. 1, pp. 12-22,
March 1969.
8. Chang, M.S.,unpublished.
9. Bai,K.J.and Yeung, R.W.,"Numerical Solutions to Free Surface Flows," Tenth Symposium on NavalHydrodynamics, June 1974.
23.
11. Harten, A., "An Efficient DifferentioIntegral Equation Technique for TimeDependent Flows with a Free Surface,"
First
International Conference on Numerical Ship
Hydrodynamics, October 1975.
12. Haussllng, N.J. and Van Eseltine, R.T.,
"A Combined Spectral
Method for Linear and Finite-Difference
Nonlinear Water Wave
Problemns,"
Naval Ship and Research Development Center Report Number 4580, 1974.
25.
26.
15
28.
29.
Baba, E., "Blunt Bow Forms and Wave Breaking," First Ship Technology and Research
Symposium, August 1975.
30.
43.
41.
42.
16
Abstract
A procedure is presented for -the
numerical calculation of wave resistance
of conventional ship forms. ave-resistance theory used in the present
calculation takes account of the nonfree-surface
linear effect on the
condition. Because of this a remarkable
attenuation of the humps and hollows of
wave resistance curve is attained in the
practical speed range of conventional
commercial ships. Taking a semisubmerged
s1 here as an example, each stage of
numerical calculations is exanined by
comparing with the analytical values.
Finally wave resistance of conventional
ship forms is calculated and compared
with experimental values. Within a
practically acceptable order of
magniude wave resistance can be
estimated by the piesent theory,
1. Introduction
............
A wave resistance theory which takes
account of the nonlinear effect on the
condition in low speeds
,Iree--surface
has been developed by Raba and Takekuima
11,2] . In this piper a procedure is
presented for the numericat calculation
of conventio.al ship forms based on this
theory.
of tramnverse-wave components.
As I
11
rr(x'y)
IU'
XY.)
0Y(X,y,D)
(4)
1
2,z
n
I
to the free-surface clev~tion
expressed by the term Cr(X,Y). Besle
the equations (1) and (,2),the radiation
teoii
diits
a
retange hod>'
'r ixed in
condition should be satisfied by P(X,Y,Z).
htrgn
oywt
ntl
system fie
It should he noted further that in the
onl the still watcr surface, we set s-axis
theory the surface- layer
present
upheara sownUind Fiaxdirected
potential d(x,y,z) does no2Zsatisfy the
n ig1
uparslocity
Ths etl
body boundary condition- The zero-order
vlciypotential isdcfincti:
Tietoa
alone Satisfies the
4(x~y~z),Otential or(x,y,Z)
Recently NewilanlIl derived
OT.(XyZ)
O(X,,7,)condition.
boundary-val1ue
same
the
ptenialindependently
the oube-bdy
for slow ships as the equation,
r is thpobebdyptniljrobulem
where ',
of tile
details
The
(1) and (2).
obtai ned from the :igid-wali problem.
derivation of the above boundary-value
is the surface-la yer velocity potential
problem and the following results are
which represents a wave motion.
found in the referencelbi.
qain
2. Outline of tlhetheaI
slw
or
512115corresponds
'1zziz1 ~
yz
~X
rid'dy'
sxy,)~d
D(',Y')
k
xy6 -/
:Surfce-ioyer
'~..
I44y..K Double-body
'/
Potential
Figl.
dik(x,y,O) ezks(X.y10),injko~~yl
Coordinate syte
0 )(x,y,Z) + o* (s.,Z)
dry
* xy) D(x,Y)
T*
Y
on 2
(x(6)us
0
sO
OX.Y
*(
(2)
CI('Y)where
aIAO
ry(XY.O)YXis
'o'
a3
TY
9/1 0"
,ko(~.gl
D~x~)rx~j-Y-0
U
where
15
AIS
~.
b,
Ta'ngc, andof integrating
integral
(12) is written:
term
the first
parts,
sec~jtile
J dxd~(xK
ee
capt~~~ep
(xeosQ i
where
It is a characteristic of the present
%ave-resistance formula that the amplitude
function is ex~sressed as at int-gral of
disturbance '(x ,yl over the free surface.
Recently MIaruo hafs derived indepen~entl\
the same wave- resistance formula a!- (9)
and (a)
vseec5
0(0vO - yss9
-d
catI
--
scg
sec'
*yin,
d0
N(x,y,O)
i
i vscc't9
(xcose + ysino)]
exi
-
nes
ji V . Cfl
se'Siuec)
ddx(xy)
seI. iveo', dyx0xy0
()
()
*yn)J,
p[iVC2
Xo0+yn)]
l~
where c i5 the cu.-ve of intersection
v~)
xyQ *N~~l
the body and the still water
0 i th
%herperurbaionveloitybetween
srae
potent iai of the double body, I e.
When a body is expressed by surface
r Usource
s~y:)
distribution o(x,y,z) and normal
doublet listribut ion p(xS),:) so as to
give zero-value )f perturbation potential
Nla,Y. )
-I0
*inside
the body, w,' have tilefollowing
Zg Ix
relations:
l
gI~
-
(0
10(x,y,O)
*O
4st
Oxv +l
-g O
I
21 0.aZ
(x,y,,O)
y,
'lx(s,y,()) onc
4n
(1
Substituting (101)
into (9), we have
(on c,
sioxA
0(x,y,Z)
1 o(x',y',z')1 j
jxyX)nr
S(4
~,Jd
T
I +x,','
!.I lS
r'
(4
weeni
h
uwr
omlt
h
whrnisteuwacomltoie
surface at the interne t ion c, and I is
thletangent. S is thlesurface of the
ui
submerged part of thc hody. Here,
body is considered which has a vertical
hutllsurface at the intersection C. MAo,
(_,l+(y2+
21
A(O)
seclo
dxi>Oxx(xi>'0)
W~e
JJdily
N(x,y,O)
41T
2
sec. a]v'e
(12)~~~V
Exclhuding The cross section between the
+ i scO(X'cosO + y'sino)]
body and the still water surface from
This we have
A(0)
AS(e) + AL(e)
AF(G)
(ax
(1S)
2
1
a2.
yZ
O(xyZ) - =y
where
a(xyz)of
sec'
_4
ASO0)
-ia
ysine))]
x exp[vsec'ef z + i(xcose
44Y e,^
+-e 2 I~ dS (x ,y, z)
n2~
Jc(-
isec28)] ,
(19)
[yo(x,y,O)cos( x,;)
sec38
sc icxy0
ALMO
+
is
S-+sin(ox,n) ]
sec2O
Zi J1
AL~a) -
1
(Tr
- v
-sec'e
JJ
-11
dxdy N(x,y,O)
+S7V
283
-"
sec e)
J2(sec e)]
Fn SOcosde
cs
cos2cos
ia 4cs
in
1
j
sec2)
sec?0),
--
(20)
(16)
where Fn - U/.ga , and J4, J1, J2 are
tne first
Bessvl term
functions
thethefirst kind.
The
of (19)of is
contribution from the surface source
It
is understood
the amplitude
function
consists that
of three
parts. One is
due to the surface singularity distri
dueto hesurac
siguariy
iti
distribution
.
doublet distribution.
It is observed
t t
he amplitude
function due to the surface
singularities AS(e) is cancelled out by
the first two terms of the amplitude
function due to the line integral term
AL(e). This fact was first pointed out
by Brard within the framework of the
linearized theory[lO].
The asymptotic expression of the sum
of AS(O) and AL() is written in the low
speed limit:
3. Wave resistance of
simple forms
XCO(
(,y
Ux
(x,y.Z) -
Ux
I
117
Sc8
O(Fn')
(17)
20
2~x
4U
T~
4U Fn'
N(x,y,0)
-a'xcontributes
_r
a.
for r - / X-'yi
8)
2a
sec'ef
A (e).4i
CW-
r sec a) dr
Ux
secle
' oe
6a
0 cos(O
7 =-F
IT6656
*Wn
r-j
Sin( I*j+
(23)
+O(Fn').
(24)
89w n'+O(n)
the wave
a comparison of cases.
Fig.3 shows curves
From
for both
resistance
this figure it is found that AF(S),
on
effect
which represents the nonlinear
the free-surface condition, plays an
important role on the attenuation of
humps and hollows of wave-resistance
curve in low-speed range.
IAF(6) -AF(S)/2a.
5io
.0
40
!L(O)
At(&)+
F=.0
2.0
43An
(22)
+ o(F )-
40
w
-4PtP'(2a) I
FG+3rTI
,lAS
1.0
G
00
-1.0
W
R~tS)~dS)'EI6)
\\~*~//~PO)
5.0
Iof
-40
-IL0
functons
f vetica cirulartwo
Pig.2. CompariS~.n of amplitude
cylinder in low speeds.
air a
Wa
224
i
C111 24FnS + 72F Fn'sin(
+
O(Fn')
(25)
On
the other obtained
hand, thcfrom
wave-resistance
the sum of
coefficient
coeficiet
otaind frm te su ofwave
three parts AS(S), AL(S) and AF(S) for
the semisubmerged sphere is given by
633C
63
4. Numerical method to
wave resistance
)cal-HuTte
in 1'
sin( -F-,
(26)
* O(a) .
A()-- isec 6
dyD(x,y)
c
- .0.-..d
c-
(phw
R.
.0
c.
F:.l t.
0i*F
L,aa4,s
m.is
'where
10
s-
Bn sin(nO)
where Fn
(,S)
F(S)
p(9)
u
W
22
()],
II/vg
cos~cose + (2E Bn slnnB)sinO
I
- p az
1- u - v
0x(X.yO)/U ,
0lZ(x,y,O) L/U
0y(XyO)/U
o
(28)
(9
cosB( T sir'2
F(B) -
.use.
1.
(29)
In the above mentioned examination of
the numerical integration, the exact
expression of F(B) is used. In practice,
however, F(6) for an arbitrary body has
to be obtained numerically.
Representing the body by finite
number of surface elements on the body,
velocity components at the null point of
each surface element can be obtained.
Taking again the semisubmerged sphere as
an example, computed values of v(S),
p(S), dp/dS, wz(B), dy/dS and F()
are
compared with the exact values at the
load waterline:
v(W)= p(S)
(
here.
af F.-0.1
"
Upwlirfiattsonueoan
t
* "
"
*o=
s
cosasin,
T
sin's,
wz()
-
= =
cos,
cosO,
9 sncosB.
C. OL_
so
I-
nl5lnti5I
-*
1.005
0.993
l.O
[
,'Q
".5
0o
Analytical
0.115927-3
0.640767-3
0.177887-2
o.t
Cw1 /Cw,
Cw,
Ew,
Numerical
0.116463-3
.635963-3
0.189489-2
number
0.15
n.20
U.2S
"
v(S)
Numerical Analytical
0.147AS5
0.27564
0.44442
0.61628
0.73920
0.96259
1.1362
1.3100
1.4839
0.14245
0.27564
0.44442
0.61628
0.78920
0.96259
1.1362
1.3100
1.4839
-0.19q96
-0.40290
-U.58736
-0.70839
-0.74859
-0.70103
-0.57045
-0.37198
-0.12913
-0.21080
-0.30283
-0.58226
-0.70751
-0.74998
-0.70340
-0.57286
-0.37370
-0.12969
0.96319
0.82567
0.57906
0.25025
-0.12401
-0.49861
-0.82839
-1.0735
-1.2040
0.95465
0.83333
0.58410
0.24830
-0.13355
-0.51539
-0.85112
-1.1004
-1.2331
dp/do
Numerical Inalytical
-0.58765
-1.2215
-1.6852
-2.0385
-2.1624
-2.0299
-1.6548
-1.0807
-0.37549
-0.63239
-1.1785
-1.7468
-2.1225
-2.2499
-2.1102
-1.7186
-1.1211
-0.38907
wz(S)
Numerical Analytical
dy/d
Numerical Analytical
F(O)
Numerical Analytical
-2.8776
-2.8769
-2.7145
-2.4527
-2.1115
-1.7031
-1.2500
-0.76229
-0.25609
0.49269
0.47999
0.45056
0.40732
0,35165
0.28527
0.21021
0.12874
0.43357-1
1.4778
1.6323
1.6980
1.6941
1.5267
1.1808
0.72631
0.29665
0.35119-1
-2.9696
-2.8868
-2.7086
-2.4481
-2.1132
-1.7142
-1.2631
0.77355
-0.26036
0.49494
0.48113
0.45143
0.40802
0.35221
0.28570
0.21052
0.1289a
0.43394-1
1.5364
!.6204
1.7313
1.7497
1.5879
1.2319
0.75816
0.30920
0.36527-1
-i-so.
Froude
number
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.2S
24
CW3
Numerical
Cw2
Cw3/Cw2
Analytical
0.111583-3
0,164280-3
0.188316-3
0.300873-3
0.406422-3
0.605154-3
0.613495-3
0.114399-2
0.146388-2
0.135996-2
0.178262-2
0.115927-3
0.169636-3
0.194107-3
0.318895-3
0.410392-3
0.640767-3
0,6326320.112825-2
0.154081-2
0.149167-2
0.177887-2
0.963
0.968
0.970
0.940
0.990
0.944
0.970
1.014
0.950
0.912
1.002
Ah
ssg7
ofwav
i.n
resstane
o
ftlil30.
-alclatin
"
s*
yj
Fg8
t.-s '------,-ragmnso
sufficint
forpracticl
accuacy uoe..
sufficientc~aigueycsse
_ .I0
form.cicl
23
27 a a 12
19
0.7764
0.7391
.966
6.770
2.s1
2.381
0.8624
6.358
2.5ar
1 23 a
5
p.,
As
}'][ 1 - () ]
aft ends.
of surfae
fo rm
0o
'|
'21
14
)5 - (
nss
il1mm
19148
({
30
1360
2330C
y.2B
si.,
AragAnso
Aom.
oe ufc
n
ted
Fi.8
ftOid1rl
to.
Finthi
~A
A I.
vausSbaie
umrcal.Frmths-
pllof
onventiona
'ff
U"t
fit'
SsA
___. _
FP
a -stneu
~~
2330C~~~~
:39
770.
2:8.27x1
0-
at Fa.20
',X1A(e)IaCoS3OeL2
05
-4
blreadthtof element
at fare and aft end$
2.0
0~Q
2.5 X03
-0.5a0x
Ia5'
1.0
--
20'
30'
40'
50*
66
- 9
ofwav spetra
17ig10.Comprisn
withO Compreisondt of v sperfa
elemt d afrt
rdt f ensurac
elemnt
fre
at
:
eds.study.
~d
4.01
2.0
k
'B()
Michell
aoi
\j
Laperient
R~o
goo
Model 1719
a1
Sm
.m Model 1720
11
as
020
130
Q10
26
1135
30at
2.0
Presen
0 955
Fa.-0.20
2X1A(G)rcosze/L!
Ta
a,
2f
Se
-
4C
- 9d ft;;ofacl
?,581 a294
2.381 Q296
0.0.5 [~oiies~03
7C.,Q-74
C.-O.86
C.-Q56
V
010~1Ol
011
8. References
[1] Baba, B.,and Takekuma, K.(1975),
A Study on Free-Surface Flow around
the Bow of Slowly Moving Full Forms,
Journal of The Society of Naval
Architects of Japan, Vol.137,1-10.
[2] Baba, E.(1975). Blunt Bow Forms and
Wave Breaking, The First STAR
Symposium, Washington, D.C.
6. Concluding remarks
In the present paper a procedure is
explained
for the calculation of wave
espitane fof ships nlclaio s
. Te
resistance
of ships in low speeds.
The
surface-layer velocity notential used in
+hepreentstuy
i
anasyptoic
1'ho present study is an asymptotic
-lution in low-speed limit. In accordance
with this fact, an asymptotic waveresistance formula in low-speed limit is
used for the calculation of wave
resistance.
relistance of
Appendix A. Wave
semi eubmerged
kee adhere
7
72 cs
os
,SALMei 16a
7F r?
r. MY
r
1 V
(XyO))-
-, 9x
)c058
for
(A-3)limit:
into (A-5), we have
Substituting
in the low-speed
-jr,
s >1 ,
Fn - UVga , r - xr +y
where
(A-2)
N~x~v,9
[-
O(Fn 3 )
W 7)
s - r/a,
16a
.- i--Fn[
9 +UF
177
20
- 13?3
MiP " U sn )
41 Fnj[(
N(xY,)
seCi8
cos30].
sins - y/a.
cosB - x/a,
) + O(FhIs).
(A-6)
X Cos(
Q
4 Fn' f ( AM
i
sece -
cos3e ] cos(
C(A-1)
=2X,+y+_Z97,2
7(Cyz
Ua jx
0(X'4yz+t-
120
)cos5]
7 cos
-
sece -
X cos(
27
cos3
) + O(Fn) .
(A-8)
in method,
the text
Substituting
into (8)
phase
and using the (A-6)
stationary
we have a wave-resistance formula of a
semisubmerged sphere when the free
surface condition is linearized:
(A-3)
Cw
Rw
TpU'(Za)l
224
Fn' + 7
Fn' sin(.2' + -T
(A-1O)
+ O(qfl.
(x,y,)}
A() + AL
(A-4)
CW
633 Pn
---
xcoO36
+ ysine )
].
(A- S)
SibstLtuting (A-Z)
+ O(F').
secelfdxdy N(x.yO)
x eip[ i Vsece
9 rw Fn' sin(
"
(A-l1)
Abstract
The method was first developed for twodimensional problems and then extended to three
dimensions. The two-dimensional work will be
described first as it is the foundation for the
three-dimensional method.
I. Mathematical Statement of theProblem
The velocity potential
following conditions:
Y2+- 0
=
*n 0
*n
++
*q
y y
inthe fluid
(1)
on the body
(2)
+2 + #2
--U ) * 0 on the
i(X
zfree
-
.0
(3)
surface
2+
gn +1
1. Introduction
Il.
30
1~t
.4;
F8
Il
-e~~
+,
l-I
F8
Figure 1.
Ux + f S(x',y')(Inr+Inr)d t
BS
*(x.y)
(5)
rdt'
_
e +.S(x',y')ln
FS
where
S(x',yl)
r is [(x-x)2+(y-y)2J
1/2
F IS [(X-XI2+(y+yl)2]112 ,C
to the image of the body).
applies
v(x.y)
where
M
SJ CY (xy)
J11
- #y
'j
rl
or
and
in the pressure.
CY (Y)d.
( - +
2
BSj r
A4 da'
r
or
f
J
31
"2
(6)
(7)
(8)
+CYNYiS
j
i
where (NX
1I,NYi) is the unit normal vector to
segment i. (Note that CXiiNX I +CYiiNYi equals
w plus the contribution from the image of
segment I for I in BS.)
On FS a four-point, upstream, finite
difference operator is used to obtain *xx so
that
xx
CD,
1
[CA1CX I 'jCi~-~
'
.1I
'+B C 'I
+CCiCXi-2,j+L'D1CXi-3,j]Sj
CBI
-,-))(xi-2
(9)
PdJ M
[CACX,J+CBCX
1
_-XI)2(x
i-li-2ilii
U2
'(xl.
2 -Xs) (xi_
3 -xi) (X,.
3
(11)
i-2i3i2
(10)
+CCCXi.2 ,j+CDiCXi_3,j]Sj-gV Si
where the four points are xi, xil ' xi-2. '1-3"
LONG REGION
+ SHORT UP STREAM HEWON
SORT DOWN STREAM REGION
fe
-4
,4
is
16
02
J
20
24
DIPOLE
INFT,
POSTllE CICULATION
/"
x'
_
.
FY -.
,
!
'
-1\ ;
Neuman
in reference [41.
.. K3ATIVE../
A,.CIRCULAION:4
an.
""
von
(5] tocalculation.
compare perturbation
with Kerczek
a nonlinear
Figure 3 methods
is a
comparlscn of wave profiles. Double-model
linesrization produced i wave more nearly In
p.ase with the nonlinear wave than did free
stream linearization. Figure 4 is a comparison
of thedrag m the vortex. For positive
vortices the doble model linearization
produced very good results. For negative
vortices the rasults were no. as goo4 but were
still much better than those for free-stream
li-tarization and were comparable to the
,
a
-z
".""."LOCATIO.
OEPT4=4.6
lENTh'12f-_v
NO IMEAR
(IALVEIN
veKE BCEK)
... RITEMIF LAII (PESIT lnlOO)
---DOUBLE
M110011LIN (RWq M[
E )
Figure 3. Wave Elevations for a
pas-turbstlon solutlmn .
Submerged Vortex
33
*1
FIRT ORDE
----DMimprovement
U
-- RIERquite
T~lED
PP.EIBITMETI'
9 PNEE STREAM WEAR
+ DOIUBLE
MODEL LINAR.
Yrface.
Fr.UMMMEN
n 1.8
16
POUTIE
ORUIT
/,C
40-
,4'
/1 +,/NUATIVEMU
SU
..
-
-EAI
-FrM
0., O
03
IAV,- 4:0
1u
2.4
WAVE
TS'U
M
S
ALIP
'1100
3,2
Figure 4. Wave
Resistance as a Function of
Vortex trengthV.
x x
Zx
2y
(2
y
Figure S. Singularity Rapresm'tetion andttv
cross sectior of the (y
34
Z 0~]9~
but 2
. +
~e~*
ey)
a(
oy
a-
) -e(x, +4o
Yy y
I ,
"
+ 0),Yy
oyy)
ot#Lt
so that
+
0g*
"
'
x)(
2 )(I-.6")
(14)
3.
mw
m
G Mg-
IMM
(15)
.75(l
Al
i
- EXPEOM
---HIN Stw
-TW
x 21 Pm LS
o 414 PA
PRESET METHO
x 286 PANELS
* 484 PANELS
.78,
1,4
o"
00
.42
.1.2
J.3
.4
panes
fo th(0x3)
fre sufac.
I;
/ex
..
.')
Te
-.3
.1
shw
VII, Conclusion
With the method presented here, It is
practical to examine the steady state
performance of ship designs. The coat is
reasonable and the accuracy of the results is
better then thet provided by thin ship theory
or by Guilloton's method [9]
9
V111. Acknowledgment
hf
EXENWT
- -COMPUTED
BY
SUrnTOETHODi
Msi
Cj_
.5.3
.3
.2~*6S.3 .31
.35
.
.3
(TaskFn
sponsored by the Naval SeaSystems Loemaond
Area SR0140301: Mathematical Sciences) and the
Navel Material Comman~d
(Task Area ZF53532001:
Logistics Te4chnology).
References
-.
.
i1.e
wi
Figure 11.
.1
.3
tIlL
.6
1.10
53
43;
4.
I *
*.6
Y3
STERN
5. Salvesen, N. and C.H. von Kerczek, "Comparison of Numerical and Perturbation Solutions
Water-Wava
Nonlinear
Two-Dimensional
of
Pobi
ms." Journal of
Ship Research,
vol. 20. No. 3. September 1976.
10.
11. Todd,
.,"Sr'lies
60-Methodical
x..periments with Models of Single-Screw
Merchant Ships." P.'vld
Taylor Model Basin
Research and Development Report 1712,
vol.13,1973.
39
aV
Abstract
A finite difference solution procedure has
been developed to study three-dimensional potential flows, both transient and steady, about
a shiplike floating body. The primary features
of this paper include the use of special coordinate transformations so that proper boundary conditions can be applied at the exact
locations of the body surface and free surface,
the application of Orlanski's numerical radiation condition to prevent unwanted wave reflections, and a new upwind-centered finitedifference scheme for numerical integration of
the time-dependent free surface conditions. The
forward motion of a shiplike floating body and
that of a submerged body are studied. Both
nonlinear and linearized free surface conditions were used for comparison purposes. The
nonlinear results were found to be significantly different frcm the linear solution for
finite-amplitude waves. Also, Orlanski's numerical radiation condition was found to be
extremely effective in reducing spurious wave
reflections.
the computer is required, and that the resulting equations are much longer and harder to
program, especially when simulating finiteamplitude waves. Consequently, this approach
is not taken in the present study. Rather, the
full nonlinear equations are used directly.
1. Introduction
*This work was sponsored by the Office of Navel Research under Contract NO0014-76-C-0455.
X - X + x cos 0 - y sin 8
The class
of body
shapes considered
here
is limited
to those
representable
by a single..
valued function on the ship's center plane.
Thus, the technique here is particularly useful
in calculating flow about a sharp-edged ship.
For blunt bodies other coordinate systems must
be used, but the formulation and computational
procedure remains the same. The presert method
can also be applied to flows about a submerged
body, or surface piercing structures, such as a
bridge pier.
I1. Governing Equations
T - t
Y- Y
x sin 0 + y cos e
Z- z
45 sin a .y
co8
(1)
OZ * Or
t +T, u~x + 9y
where
- u00 +wy
v -
vO - WX
(2)
OX
Y
'Y,Z.T
while
_____
r
--
Iy,Z,t
'.
-5-
hXX
YY
*zz " 0,
()
4)
In E4. (4, P
the fluid pressure and p the
density. Using Eqs.(1).Eqs. (3) and (4)can
be written as
X,
X
xx
and
40
mv.
#t+
2. 0
p
+ 9z+
(6)
(vB-;VF).Vtn '
~x~
n 2
=
.f
(0
Z'z
t,- t.
Let Q be any scaler quantity, then by the chein
rule we have
(B)
03
xu
Q
Q2 .
(11)
-z
I/AUsing
1/Aequation
z/A
oxx
+ 11 +
(f
d2]oy~y_
z)
+ ozz_
where
A - J1 + (fx)T
+(f
y(~~ z~
z)T .
(fxx +
+ y,~ -
*y - q
0
o
(12)
z)
(9)
X Yx
2xy
+(
) 2+
+ 0(13)
gz,+ P
The hullsurface boundary condition, Eq.(9).
now takes the form
~YI
2
f(O + #K) - V+f1.1 ]/[ +(f5 ) + (fI),].
(14)
Canter Plane
AV
0s.
(a)
z-
Top View
rFREE SURFACE
FREE SURFACE
0 .
+f
Eqs.(1)and (11),
Applying the
totransformations
this leads
"+(+xnx +
y
t
j
bove
te
the free
of quntites
displacement
theeuatins
which governssurfce.In
In the equations above, quantities
like
, $a. etc. are to be evaluated by
Eqs. tl)!YThe
Coordinate System for Free Surface Conditions
To impose correctly the free surface conditions for finite-amplitude waves, it is im-
(20)
-
fzy.
V+
x"
+ 0. + x)nx
""
- z
(22)
Z- = Z' - n(x',yt)
t" - t'.
(17)
(1+ fzny-)*z -
zz
fx(O
Ot"
-- - ny--*z--
"
OY_"
x'
y=
(15)
Note that in Eqs.(18)-(20) all space derivaties can be evaluated on the plane z" - 0,
the only exception being z- which Involves
Since
variation of * below the freesurface.
ced by
esy computed
cb bee easily
o #z- can
by
varitio
z,, " #2',.
using the (x',y',z',t') system which describes
the interior flow field.
hullsurface condition, Eq. (14).
becomes
(16)
+ x)fx - #zfjny - az
Ox
1X5)
"(x
1
(18)
+.
+ pg'0
t (u + C'x)nx
- *zfzny-
+imposed
v + Cy " (u + x~fx
$z(19)
where
x
+ r
+ 2n -(f z
+I+
The basic computational mesh is rectangular in the (x',y',z') space, as shown In Figs.
3(a)-(c). The only quantities to be computed
system are n and 0 at the
in the (x-oy-,z-)
free surface. The following Is a sketch of the
steps required td advance the flow field In
tim.
+ 2)2
f2 fzn..Z
fxnx.,.)]02_1
These equations are to be used at the free surface where z' - n (or z- - 0). Thus, the
term gz' has been replaced by gn In Eq. (18).
The following relations are used to evaluate
*t' and #a,which appear in the equations
surface.
ebOv.
43
0+1
1,1
let us assume
For illustrative purposes,
constant mesh spacings 6x', 6y',and 6z'. It
wasfound early in this study that the computation is unstable whenthe familiar scheme of
explicit central difference is emeploycd. Since
in forward motion, the flow clearly has a preferred direction, one imediately thinks of the
is
Unfortunately,
one-sided, upwind difference.
the accuracy
damping that
tMe usualupwind difference
schemes introduce
so much artificial
seriously impaired. In this study we use a new
upwind-centered scheme which introduces no
artificial viscosity for pure advections at a
constant velocity.
Let Q be any quantity, such as 0 or n, defined at the free surface. Weuse the indices
(i,jn) to discretize Q over the free surface,
such that
=
y" , n 6t" ) ,
Q , j Q(i x" , j
where 6x'" and 6y" are the spatial mesh spacings and St" is the time increment. Equations
(18) and (19)can be cast in the general form
Qt"
U*Qx."
v*Qy_ + t - 0
=Q
[ .]. {
0l-,1
aX"
(a) x"-t" Plane
.) , .
.j)
[Q]"
(O +
n-
0x)f
C)",
,"P on.
(b, x"-
,J-I + Oi-tj+l
/(26y-)
where
QA'-
T
al"
Figure 4.
[Qy10 "
.- ii+iT
,1+I
QB)/6t"
-<
i0
(23)
1,
,
QI.r
I+ n,,)
[a
i~
Q7:1,J).
(24)
1'4
In Fig.
nimposing
[Ox']jn
[Or~ji1,
ijk +
(SxT)-2
'OY'f,j,k
,ny
fiJ+l,k " 2?,k
(,y-)-2
(n
z,j,k
4i,j,k+1
* (6Z1
nn
+ ti,j-Ikj
n
+ n
ijk.J'C
'i,j~k
j,k-11
k
"2
nn
,j,k
n
t+l,J-!,k
0
CQX = a
(27)
-lj+lk
)
n
01
-lJ-l,k)
* (4 6x' dy^)1
n
1Oy'zli,jk
k
~yz{4~ IOi,j4-1,k+l
- j
i,J-1,k+l
of~
+(
(28)
i-}
(8
+ n
M
" l,j+l,k-1
i~j-l,k-Il1
c6t
(4 6y' SzT
lYLjk
I(Ii,+k
nnThe
J-I1' I( 6y'
= 0
1.
By
inserting
component
in of
Eq.
(28),it a
canFourier
be shown
that thesolution
amplitude
wave components of all lengths is preserved
under the algebraic operations in Eq. (28).
(26)
in Eq. (12)and solving the result ingsystem of
difference equations by the standard procedure
of successive over-relaxation (SOR), subject to
appropriate boundary conlitions. Note that
Sx " nd dy
6dy"in Eqs.(26). At the
hullsurface, Eq. (14) is represented by a onesided difference in the y'-direction, while
fn
"(
B.
-
,,i
+n-
Qn-1)(
Ie-
"-2
B-I
B-2 (B-2
(29)
(30)
Wheqn
a wave crest or trough approaches the outflow boundary. Eq. (30)can result in division
Of ze by zero, a singular situation that must
be treated separately. Thus, when the absolute
value of the denominator In Eq. (29) becomes
very small, e.g.,
45
rv
"
LENGTH OF SHIP
wher-eS1TOTAL
1x
NOMNAL LENOTH
OFSHIP
D
i
0IF I
nIB-0.o
h-L
'
-0.4
(31)
(31)
04 '
0.4
'S
0.6
) TopV*Sw
oo
if a* <0
(0
-
eBOW
SERN
if 0 S*S 1
(32)
if a,> 1 03
In the present study, Eqs. (28)-(32) are
applied to * in the interior near the outflow
plane of the computation domain. 1heeffectivenessof this treatment willbe discussed in the
next section.
-O a
(33)
Ot,+ U*0" + gn + P- 0
o.2
0.1
(34)
where u* - -ue* - (Xc)t and gz' has been replaced by gn. Equations (33) and (34)are to
be applied at the initial, undisturbed position
of the free surface.
.4
14.
The
tion
o.i 0
.2
SeIlino,
(b) Cross
Figure 5. Shiplike Body Used in Sample
Calculations
,o (Xc)t{
(uO)mX (t/tmax) . 0
(Uo)M
nt
co
x ,
t < mx
It . ax
(35)
Figures 6(a) and (b) compare the free surface profiles along y' - 0 (Fig. 3(a)) between
linear and nonlinear calculations for
(Uo)mx * -1.0 and tm - 4.0. Figure 6(a)
n the ship Is still
corresponds to t 2.5n
accelerating, and the flow is unsteady. At
t - 10.0 (Fig. 6(b)) the flow in the computatian domin hs reached steady state. At
" 1.0.
steady state. the Froude number IuOI
The nonlinear result is quite different frim
the linear one. something to be expected because in this case ka w 0.65 for the linear
solution (k is the wave ,nuer and a the amplitude), The wavepattern at three different
tims for the nonlinear calculation is shown
In Fig . 7(a)-(c) in term of contour plots of n.
The mthod of this aper is also
ble to a sumard body. As a nuieric
experiment, both thiner and nonlinear calculatioms tie made for the forward motion of a
body with nondimeasional length equal to 1.0,
and a dimeter of 0.2. The depth of submergence is 0.58. The motion is again prescribed
by Eq. (35),with (uo)x - - 1.0 and tax *
4.0. lte nonlinear and linear free surface
profiles at y" - 0 are compared in Figs. 8(a)
end (b). Note thatthe profiles in Fig. 8(b)
are in steddy state. In this submerged case,
as qposrd to the surface ship above, the
source of disturbance is farther from the free
surface., and the linearized free surface conditions are exiectd to be good approximations,
as is avide't in these comparisons. In this
case, ha - 0.1. The contour plots of the free
surface are shown in Figs.9(a)-(c) for three
different times. The flow field in the compu4;
I,'nhas reached steady state in
Fig.9(c).
In the exaples abeve, the body na:mnitially at rest and then accelerated to a firal
,
constant velocity. In due time the flow field
in the computation domain should reach a steady
state. It is interesting to see how the steady
state is approached in different parts of the
field. Furthermore, it is a nontrivial question whether the sa. steady-state solutien is
obtained if the s.i- elerates in different
manners before re' t
he prescribed final
velocity, because thi: .w is nonlinear and it
is -3tclear how the nemerical radiation condition and truncation errors in general affect
the well-posedness of the overall problem. To
answer thisquestion, an additional calculation
was made for the nonlinear problem in Figs. 6,
this time theship being impulsively set into
forward motion with uo - - 1.0 in one time
step. Figure 10(a)compares the steady-state
free surface profile at y' - 0 between the
gradually started case (i.e.,the problem associated with Figs. 6) and the impulsively started one. The agreement is excellent except for
minor discrepancy at the last few points. Note
that the last point (point C) is subject to
numerical radiation condition. This kind of
agreement is observed throughout the entire
computation domain. The timehistory of n at
three selected points A, B, and C (see Fig.
10(a)for definitions) is compared in Fig.
10(b). It is seen that, independent of starting conditions, a unique steady state is approached, except for minor variations at
points very close to the outflow boundary.
References
1. Chan, R. K.-C. and J. S. Stuhmiller,
"Numerical Solution of Unsteady Ship Wave
Problems," Eleventh Symposium on Naval
Hydrodynamics, Lonaon, 1976.
2. Orlanski, I., "A Simple Boundary Condition
for Unbounded Hyperbolic Flows," Journal
of Computational Physics, 21, 1976.
3. Stoker, J. J.,Water Waves, Interscience
"
Publishers, lnc.7iRw o-r , 1966.
47
IT
TTLLENGTH OF SHIP
OMINAL LENGThI
OF SHIP
7 0.
NO L N A
(o)
0.
t=2.5 (TRANSIENT)
.01.
-10
1x
b0 t
(c) t
0,
*15.0
0.5
1Tran.5ent)
(Steady State)
IEGT
FSUdMERGEO 8
0.04r00000
-1.0~~
-00
1.2RA(Tnsient t. 3=
(a)
0.04
0 052.
15.0
0
0-yo
00
77ue9
reSrae
09_
otu
asfrte
rbe
nFgr
GRADUALLY STARTED
IMPULSIVELY STARTED
0.10C
-1.0
-0.5
0.5
1.0
1.5
--
0.10
i7'
08
GRADUALLY STARTED
IMPULS.VELY STARTED
2.0
T0A
LENGTH---FSHIP
0.
It-OUTFLOW BOUNDARY
OF THE SHORTDOMAIN
_NOMINAL
0
-
SHORTDOMAIN
LONG DOMAIN
-1.0 1
(a)
1.0~
0.8
-- TOTA SL~H
1
NOMINAL
06
II
OUTFLOW
BOUNDARY
LE NGTH
OF SHIP
I
0.41
0.2
-I0
Figure 11.
I2
0.20~
0O'5 -
0.0
0.05-
00
-1.5
-1.0
-0.5
STEADY
STATE
-0.10
005L
(a)
0 5
TOTAL LENGTH OF SHI P
0.40
3j-
OM INAL LENG IH
00
00
00
-0 1.5
-.
-0.2 a
0500510.
SEADYSTAT
U4l12
Fi0ur
at o of t e A eq a y o
0
in a
0
p ro i a i n
DISCUSSIONS
Ofthree ppers
Invited Discussion
.NM Newman
We should appreciate especially the comparisons for tie common Wigley model that have been
added by Drs. Dawson and Chaa following Dr.
Babe's introduction of this hull form in his
preliminary summary. There is clearly great
value in the use of common hull forms ini
comparing different numerical techniques, as in the
analogous towing-tank tests fostered by theITTC.
The paper by Dr. Dawson includes calculations of thefree-surface elevation and of the
wave resistance; the crucial step of one-sided
finite-differencing appears to embrace the
radiation condition as a part of thefree-surface
condition. From theanalytic viewpoint this seems
a marriage of convenience, but the results are
impressive. One detailed comment Is that the
double-hull linearized free-surface condition
differs from that derived by Dr. Babaand myself;
Dr. Dawion applies the free-surface boundary
condition on zO, as opposed to the "double-body
free surface", with differences that appear to
be of leading order. Itwould seemno more difficult touse the more complete free-surface
condition equation (2)of Drs. Babe and Hare.
However the comparison in wave-resistance calculations between these two papers isstriking and
63
Discussion
Fy H.-Maruo
of paper by E. Babe and
M. Hara
*(P)
So,
,()
o1
SQ
Lo. 3
o,0,.,,.
./3
/
o----------
+ ff
(1)
f/
/
.00
*(,y.)G(PQ)I,d'dy'
/G(PQ)
_n
o.
40
+
2
y)
k(2u + U + V
32
030a030
Ov
a++v (U
W2
(2)
zcorresponds
z-O
integration.
00
+ V
+w)
(JJIn-IA)exp[kz + k(xcose
S
*(x,y)exp
k(xcose + ysine
dxdy
to
Is
(3)
on
on F
They are shown in Fig.2. The result including the line-Integral and the free surface
/2sources
I
33
(4)
54
Re
Il
y
a and M. Hara
to discussion by C.M. Lee
There are three reasons why we do not use
the wave-cut results for the comparison with
theoretical results.
(1) The wave-cut results do not always give the
wave resistance. The term "wave resistance"
used here is defined as the resistance component due to thegeneration of gravity waves.
For conventional ship forms, this component
is written as the sum of two components:
Wave Resistance - resistance component due
to the generation of
propagating gravlty waves
+ resistance component due
to wave bred!ing
(2) The measured wave heights are the results
of wave-wci interaction. The wave-cut results
contain interaction effects between the propagating waves and the viscous wake. Therefore, the
wave-cut results do not always represent the
resistance component due to the propagating
gravit) 4avesas was pointed out by Prof. Landweber in the keynote address.
(3) The process of analysing the measured wave
heiqhts is not always accurate enough. The
theory used in the analysis Is the linearized
theory for an idealflow.
However, the comparison with the wave-cut
results is useful for the investigation into
the detailed mechanism of wave propagation. The
order of magnitude of wave-cut results is about
80% of the wave resistance defined by the Hughes
method for fine ship forms such as a high speed
container ship. For fullforms, the wave-cut
results are much less than the values given by
the Hughes method. There are two reasons for
this. One is due to wave breaking and the other
is due to wave-wake interaction.
The theoretical value of wave resistance is
originally defined as the resistance component
due to wave generation. Therefore, in the
present paper, we considered that, within the
limitation of our present knowledge of the
resistance components, the calculated wave
resistance should be compared with the results
obtained from a method such as the one by
Hughes where the form effect is considered.
Author's Reply
byRobert K. C. Chan
Professor asNewman's
my the
paperradiamay
be summarized
(1) thatcomments
the needon for
tion condition in an initial-value problem was
unexpected, (2) a single radiation condition was
imposed, instead of two separate ones, and
(3) that te calculated wavelength seems much
shorter than expected. To answer Professor
Newman's first question, I must admit that there
is confusion of terminol
. In the usual
analytic approach to initial-value waveresistance problems, an infinite domain is
assumed where the only far--fiecondition is
_I55
as
mhowever,
that even in the case of a body kinematically equivalent to a distribution of singularities, the integral equation used to calculate the density of these singularities may
have a number of solutions for some, so-called,
irregular frequencies (28).
Contents
I.
2.
3.
4.
Introduction.
Notations.
Hain results.
The flat bump approach, I generalized
solution,
5. The flat bump approach, II strict
solution,
6. Two auxiliary boundary value problems.
7. Existence, uniqueness and regularity of
the solution : proofs of theorem I and II
8. The I 'hod of Singularitius : proof of
teorem III.
Referenon
(s)
I. Introduction
frequently, numerical methods in ship hydrodynamics are developed before it has been
proved that an exact solution to the problem
really exists. Prom a practical point of view,
one may maintain thih procedure when good numerical evaluations can be made which compare
reasonably well with experimental results. For
example, the Method of Sigularities has been
applied successfully to calculate the difiraction and vagiatlon forces acting upon a body in
heaving motion in waves. Nevertheless. even in
t?
57I
it is clear that the non-uniqueness of the solution in the hypothesis of the zero Froude numher does not necessarily entail an analogous
property for the corresponding solution with an
exact Froude number.
The above discussion demonstrates the difficulty presented by the study of the existence
and uniqueness of the solution to the N-K
problem. When the body pierces the free surface
the mathematical problems are enormous. To begin with, it seems to be preferable to study
the simplest case of inersed bodies and especially of bodies lying on the bottom of the sea
or 3-D bump) and subjected to the effect
of a flow of constant and uniform velocity c.
The potential '(X) corresponding to the flow
over this bump is solution of a N-K problem,
which shall be noted as NKd (Fb, E) ; d is the
of the physical space (d - 2 or 3)
K(X,) o (Y) dy
](fr
The flow condition on r is
(.)
OCX)
x(2-D
Sv
(1.2)
v(x) ,
x c r
where N is the normal towards the outside relarive to thc body end v(X) is a given function,
t.i and (.2) give
( ) .
gdimension
(1.3) O (X)
O(Y) W
CI.Y)dvr - v(X)
Fh - c/g is the Froude number allied to the
r
depth h, E - C(X') is the height of the bump
which is an integral equation of the second
at point V , positively counted from the bottom.
kind for the unknown a (.).Let V be the integral
operator
To prove the existence and uniqueness, the
0.) r
R
(.Y) dK
NKd (Fh, T) problem shall be considered as an
a
f'r"~ ' x
Y
elliptic boundary value problem (but with a
the equation (1.3) can also be written as :non-conventional boundary condition at the free
th
(
3 cn asurface)
and the perturbation theory of linear
(1.5)
(V -
)o - v
(U)l1 is
a :ontinuous func-
2. Notations
5.
(2.2)
(2.3)
ma~
X' K
1(l)(X')
E(X')~
E(X')
6'
E
(2.4) O~(w)
t-, t(V)
(2.5) &()u~~
(2.5-1 N'~w)
x-)For
vatives up to the a-tb order (m<-) on th, comnpact subsets of w (of w) If w is compact
V6T).
em (UW
0
UE2Y CO) iff UE Pi(w) has compact support,
supp u,.i w.
N)U
(X)- ) iff u23m(w) for all m 2! 0.
the topologies of theae spaces sae (31)
jndu anshs t n
u.,mQd-')iff u' 1C
finity
,n du.$K,' CRd ), wwcee K is a compactum ofRN
u
'a (ft
d1) and supp u cK. The Banach
am,0 CIRd-1 is a Banach Algebra, that is i
sTpce K
Cd-1).a uvc~a,0 ORd-I) and
OR~~?~
(2.11) 1u 1 .am
"
d-I
K1 ) v
~C
3d-,
N
0<1t
('.).
3 h - t10)
The fluid domain is therefore in terms of parsmeter C.R
O~d-I)I
K ~
~..o ;u;'m,a
1
v ;1U.0 Ckd-I)II
0
iff u,
'u(Q,)and u Roe
u~j"(2)
(P:
In patcua
(28)%
1..()*{XY.
X3
2d-1
d_1
(InIJX')7
,-if X> 0,
(2.12)It(X)2d-2
I(+X
0<3 .h)
Functional spaces
The termipology and the notezions are for the
most part those of L.Schwartz (31). Let wbe an
2
open subset of in, n 1bdry w, the boundary of
w and wT wubdry w. A function u(X). Xe w , belong* to the class Cm in w (in w;) iff u has con-K
tinuous partial derivatives up to the rn-tb
order
in w (in ii).It is said that u belongs to the
clogs Cm,Llin Ziiff u belongs to the class Cmt
of the
in wiand if the partial derivatives 3 "
(0<u
rn-tborder, are a-hiilderian
0
, Ythe
au
(Y
jau(X)I _SMI XYja
(2.9
(2.9
3'~(Y)II4
I~a
I-Y1 XYnenta
1x3
where M is X'y inde andent and where
Euclidian norm snR.
.Iis the
~.
JThe
in
3. Main results
It should be remembered that QI is the fluid
domain defined by (2.7), N is thecnormal to
bottom, with (N1 ,N2 ,t!3 ) or (N1 ,N3) compowith respect to 0K1 X2 X3 or OX IX3, OMX
is the potential at point X allied to the fluid
movement of v'lichthe velocity is
apumI .Ijx-YI
";
XY.w)
r(3,~-
8'~i
d.
03)
according to d - 3 or 2,
E(x) be the height of the bump at
point XV. We define
')I~~
Iju
norm is
~0X
31
3X
with
(3.2) V'(,,
(33
$up {lpumX
311.
f$
Cd)
m'2 it halb sidht
adta
?2
thllb
Kn
the function u(X) defined on rbelongs to
1
510(r) iff u(X',h- E()e1 ,"'Rd-1).
ucla"'~ (w-)
1ff u is a function of the class
CON
(l
EM
0()
.*(
EM (XI)
N,(')(X'
r EM
N,(' -IIo~
by formulae (2.?)
to (2.5).
where the 6utmatin E1* taken for all derivatives up to the rn-tborder.
Al pcsaoeare
h
banach sae.which
All he saresabov
spaes.chosen
uvCOw) iff u is a function of thetlaosC low
u.C(a)
1ff u is a function of the class
in w
uot(w) iff %efCOMs for all a.0
us &W() 1ff uoee6)
These four spaces are equipped with the topo logy of uniform Conergence of u and itsderi-
as theunknown N'nction.
NK (Th. c , I~~
find $0I)
problem
$0() (x j C ,
(NK1 )
V22
(X ;
(NK2 )
2 X_(-)
ko a(l)
I.
(K 3 )
, XT
O
Theorem 3
Under the conditions of theorem 2,
the ,nique solution of the NKd (Ph,E
) problem
can be expressed equally well in any of the three
following different forms
X3 - 0
)W.7
a
(Pc)
(V - II) a - v,
here k. - -L-
(ii)
C2
Given
(D - )
= u, u - restriction of (cXI) tOr
(iii) mixed potential:A(X) - Ihi- v where p is
the bd1p(e)
K compactumo ofof
m integer >=2
at real 0 < :the
a <
We net
(3.
' (space
unique solution in
integral equation
of solutions
M(i))
(D -
of data
I
K
(ae
d
The two folloing results are proven
Notations
I~
(3.6) B a - f
-
K (X,Y) a (y) d r
(X,'Y) Pi () dy [1r~ ,
Sv.
(3/
1l,o(r) of the
m 'a
(3.5) W
v A c NI
Ir
and he restrictions of So and Dv to the surface rot the bump are noted
as So and D1 . The
followinged
drf
JaK
r.)Vo-(xMY)
a (Y) dy[r.
- (x
#o(
so
0 1 X) I h) so that i7 . Rd-I 9 T
f(XJ.X2X
3)
to the mapping Xo
f(XoX2'd3)
()(i.t(I))4olwtion of the
u, ZPFOd-1
(rd-
l)
d-i
uc 'D (It
u.
(dR
u is a function of the
class C
(iiuand all iopeeiva'a2tor
(ii) u and all io derivaties belong to LP(Rd )
iff Mi
So
u is a function of che
,
class
(Ui' u is a fuction slculv
mncreasin 5 at irsL-iy.
R -13 ift (i
I)
S'oRd'.
onto
0, OR I) space of "stributions
lio St intinity.
'
rapidly facrea-
D
ox3
(4.6) B2(A3)3 o
d-1
!'RI
(30 p.49).
d- l
S' OR
Inverae Furier
Ttasfnofo((Pt).
2W 'IL
fi(k',k
3 )-
(YXi)eik'
d'
if d-2 ar,&
-)R
d-I
(4.2) f(X',X1)
-(2,)
-r
(IF)
d(X) - ( in 13
(G2)
kd2
..
L
SN~d(Fh;O,I('))
(4.1)
of
is
-(k
- 0
I
(0i)
6(X :
X3
X' - L.
Vnc
1h.
if
yo increa-
H2 (kX
22 >2 h2
k oh-h2
3)
Pf
-
2 (xt.x3 )
h2 (k,X 3 )
Y-t (112(k,X3))
'
with
j-O-t- - The 2positive solution y
.emma 4.2
th yh .t
t [0, 2. ]
y(t)of te
h2(kX
-k k2ckhk
0 (chkX 3
k2(hh
ko
hkX)
'h k)
k-)
(ii)
( y (7-0)-y~t)
Y (-t) - y(t)
(Vi)
if Ph 0 I 93 (X) - as+aiX I
+
k
2
k
with k I
AMto T
t
) increases from 0 to+
>
0,
each Fh,y' (t)M
2
3gm 2t0
1koh hy(t)
H 3 (X) - Y
2(X)
8-.)7
kh
if Ph-I
if
I t2 (X)(ch ,X l -
Irn
( 3(k
k,. X 3 )}
' 63 (kI'
k2' X3 ))
-2
XI)
Oc ORd ) Then the NKd(h,0 :(' ) problem has an infinity of genorelised solutions
given by :
10(It)
Mn- E-D,4
(8')
0o(X)+
X
, 13d(X 3)+
'
(X)
where
Gd -is any
one of (the
ganeralized Green
potentials of theorem 4.1 and where id is an
rv olynomial, that -an be chosen indopendently of Gd but havtng tho mae fom ae the
8d polynomiale defined ir theorem ;. :,
The i:mbol
X','
denote@ the convolution with
respct to variable X'.
28)
1 + 2 (X-X2hX
3
+63(X?-3IXXI+6hX1 X3 )
I
ko sh k X3 ) TC
aI
I g
if Ph
cons-
Theorem 4.' -
There exists an infinite number of generalimed (;ronpotentials that are given by the
following
a :
Go oXn o- Hd(X)) :d(
+
Cd(X)
+ Pd(X) 0 gd(X)
where Hd, Id and gd a'o given by the follwingi
foimiulae :arbitr
(i) in the cas8 of .7
.
X3 -
. hkk
Ph 0 I S2(X)
ada, abitr
Theorem 4.1
if
, a
shkX 3 )
)tchkh-ko
ThhT
extension
to Rt of an arbitrary distribution
C whose support is in (C) (see lenma 4.3(x))
and h y'(t)-
/ le + k
at2m
3 (k 1 ,k 2.X 3 )(ch
I~
zn ohich case
t-O.
(viii.)
if Ph > I and if t-to,
h y(t)= /30'-
-k(chkX 3 - kk
(ix)
a
So+a
1 X, 3+ a2 X2 I a 3 XX 2
3Xta(X
3
6hX3 )XM
3
2
H3 (kl,k 2 ,X3 ) - Pf h3 (k,,k ,X )
2
3
for
a 2 X2 + a3XIX2
if Ph - I g3 (X)
-
12(8)- 0
9 AsYX3)'
Let us met
0 (k',83) -
where k
62
Ib'
(-kI
if d-2)
ke
khi
- ehkX 3
C (t,SRd-)
(a) D U-0
X r)o
2
0
(b) D X U - ko Dx U
(ii)
case d-2 - The same operation is carried
out and nstead of (4.9) one obtains
'
0(k)
)P .X[o(k
3
( h
k-k--3k)+a 2 2
(k).a3 2
](3)
k X3 fk)
)+e )MGk'X
(2) (kb).
6(1)+ (1)(
+and-,()()2
+k
(4.10A)
,X
]
D ,OX
k2 (3 j 2
I3'] vkl)
1Vw
(i.)
'(k')
I
.0 (k',X3 )-Pff -k
[;2
d(1
6(k)
- . a
0(k' K)h
) chkh'ko! - lJ
DS(k')
2
k')
D 6(k)
I-/''2
Study of TI distribution
a m
*.'-"
imwi,
0' I k2 J'
D
where TC is a single layer on (C) (i.e. a distributlon with support in (C)) and TC is the axtension of this distribution to the (k)'k) 2
(31. theor. XXXVI p.10 , formula
pl'cs - sea1 14
(1V,5;7) p.
and theor. VIll p.2) , The
terms ai and a
are arbitraty constants and
aem, i~ h 0
1k cos I, B sin
4
4
where 0(k k)
-Ik
(4.13)
seudo-function
(C) is here reduced to two
The manifold
points and the meaning of TC is therefore
elementary.
By taking the IFT of (4.10A) the formulae of
(4.10)
O(k
o-
1Xk')
where
The FT of (4.6) is
(k"X)
B2(X3)
where
)
2
th 3 )i (k,t)kdk
(t.) 3 (kr.X
f+
o
is the finite pert of the integral.
f
what follows2 it is sufficient
the elements of Sin
u h that
. OFor
8'
63
to consir
(4.14)
j(k,t)
(S2),
8
ly
wA
Theorem 4.3
When Fh< I, Io
f[)
]0,
w whioh follows
n
<.,.> is the duality pairi
If X _ h tjo above formu,
if f14OR ).
~ 2X)ideie772
I
kcoa2tchkh-k
<Tc' *(k'-
2 )>-
<Tt.
ahkhj
Y()chY
2 (X)
(4.16)< YI
")
41f
i(y(t),t)>
,t
),t
In other words TC has the form
Yt
~h
'
~~Tr
chyt(X
3 -h)
h.,
Cos
2 .oh
1 (t) and
and Mt A t rcon
t-)
CtTh
exp~not]y(t)
-w
ch
_2
t
(.)(kkX
ch[ty(t)(X3=
3
2'3c
-h[y(t)h
l ch
c
ch
Y3(1)-(2n)j
h.
73X')
IIJ-
i-f
dii(t)
3r<
ha i(b
. Adan masure
~isxste.c
,
(bcosX,+bs
o
2 s noX
cnYoh
if Fh<l
cp
of the integral.
if Ph Z I
.
where it is assumed that t varies in
Knowing 1he properties of y(t) (lessa 4.2)
it is easy to prove that *[Y( t)I
8 OR
is continous function of t and is bounded
don measure ie.
,Thuo T isae
in - ,
~~
'inite par
denotes the
'i
2 (X) - 0
(iv)
W) If h< I
X
-0 Ilx
2(X'X
O2 (koX 3 )
G
)j=2
d v(t)~
wit+
h~h
0 (K+ h) .ntyoX
h ChY("l'"-
x)
(x
+ x2 )
)sgY1 + 1'2 (X)
th
" (X1 ,X2 )i(rcoOrsinO)d
s - y(t),X'
B-.
k2(chkh- A h . kh'.. dk
k2(chkh-r shkh)
IT
o
k
ik'X'
dk
hkx e
chkXa
I
dtj k X3 z hk h 31 e " k
-)
k
k,
H 3 (X)--
h
here
on 'it )xSoR ).
ataine its meaning
where
on the
(4.15)
O , and
free surface X3 - 0
H2 (X)--
2
Definition 4.3 - When Ph < I the pseudo-znoton
* S oR )by:
Pfh 3 (k lk2 X3 ) i s de f nd for ea h
1A
I+(t, )dt where
<Pfh (k ,. ,X ), *(k 1 k2)
~~
(ii)
(ts
alue of Ph.
any ge--
(i)
Fropoaition 4.. -
that
(k,1w
- t)
then to prove that
-j(k,-.t)-
It is not difficult
3(X
.Y(t) co
-h(2sff
,dv(t
-2
hR)
- sin Zt dj 2 (t)}
X1
3.
(i tI)
!
X ((Z) 2
* 6 2 () * V2 (X) + X2")
with x2 (X1.X3 ). f O)j
64
0,CR)
-().
I
(C2) if Fh I
G2 (XI,X 3 )-
with
x2(xlx
k
k
W(A)if Ph < I
chyt(Z
3 -h)
2
......
tion 4.
-in2Fdt
2
+ g3(X) + '3(X) + X3 (X)
r cos (t-6)
X3 (XX
2 \ {0,01)
3r
(B3) if Fh _!I
a2 3
-- (x',X
3)-
IT -to0]cy(X3 h)
TIT
f
[ 0
'
(5.0)O6(X
1,
Let us set :
(5.3)
e--iti
V, it'. o
o)
SO'
.2)
452
so that
(5.4)
ht!O
kI E()(X_
()
k 3E
"
d!
h-kdXI c
2
.+ 3'
c [3mo:Xl6m Xl
] SgX I
XI
o(xi.x 3)- -
(5.1)is o(I).
(X)
~~
no
... . i
(X) 02 (XI)
prnlast term of
becaus he
d-
(X)
_IlHO)
I)Y
x - -l
X i3arcularXlm
a tCt2aCMe-,2
1
-3 3cXlmo+3a3c 3 o-6hX3 oa3C+ 2(X)+o(1)
swhere
at vseof
8 he
X-O o
o
S )and
&(j,
I
E01" 'IO~d -ao),
m
(o,(
o
u a
its
~eavrae
that fe,
2X3
(h
aE
1I(Xi) 02
t4he
others,
~~~~
(
'i0dl
3
(I )
: |~~
~Mh
N* 1
S. The Flat-BuMA
C.
CIL
o~')'o
d
Xj
+ c
O \(0
X3 (X',X 3 ))
X3 ' 0 ( wr
when r
!e
)
+ 3cX3
k(h
3(X) + X3 (X)
+ g3 (X) +
3 )_
c
V
zh2 t
hth-4
( )
-l
3 )*
(XX
y'(tsinj'j' le
Ch-f h
G 3 (X'.X3 )-
that 0
To prove
k /
(NO
to
0'1 ()
g(K*)
3)
3
(-5)a :'~a
x
3 t1
3c
2
7
testoo- 3a3%loe
a:d
-1a,,XjgX
|
4h+
2
Yl the
2
_1
(g) d- 3
According to theorem (4.4) the generalized
solution is written as
z(I)
(5.12) 0 (X',X 3 )-c N-1 (X
3 )(X,)I3 ( '.X
3)
1(I)
Td
(X'X
).
+c
with
,/tchy
(5.6) %o(X,.X 3 )
(5.13) 1
--
()
(I
d*9(l+cIX1
2__0
thy
Cosyrco(t
_______
n
2(x1)
(I)- E(I) i
c "
|e);
T. oo(ho3
dX
+
*
(5.14) c (
g (XI)
(X',,
X-/ 3 )' 93 (X') - - c mn.
hy.X
3 -
K
fz
(5-15) m
SgX1+amoC+1+OXl~
sh X3)
11
1S
11
h 2The
dX'
W(x')
x2 E
+
b
(c
1(I)+ b2
EMl)
0
a-o-0 and a1
()
(
Io -values
I) CBo
S
0
(I
t ) But
bl
7T3 (V)
a0 + I XI + 2 2
arbitrary signed measures p,1and 02nd
the arbitrary constants at. must now be derived
J
for (5.12) to satisfy WK4 It .mmediatly follows
that :
2
0
. , c, b2- 0
B
a- 2
(t) ,'
. aco
c ao0 a
0
0.
(t)
c 50
aX
II
fourth term equals
term
= isO (-),the
third
T)cosYoXl]
the
3
hence.
k
o h
h
1i
with notation
k~o
ho 3)
shyoX)
c hyoX
['S IinyX1+E0c.X
1
(b
d(
x )
C I -0
S
almoc
k ro
,N3-n)
x3)
(t)
2(xi,)
X
3
+ c TxT ( Ii)
83')
(X.
0
So
),)
X
o(I)
Y'(t)
(t)
)
.- 2-t (-T
e
hence the behaviour at infinity Xl
constant
ia
2
(S.J6) 0()
- c
"
(x') (1,)
to
chy(t3 -h)
(t)
dtl
,
tIty -Wtth
hnd 1I) steddy phea* method shows that. (5.16)is
0(r'' )wh ch satisfies NK4 condition.
.
d-2,
I and
>1 Ph
andi0B
if Ph
vanishedthat
Bo is
be assumed
Let It
o equale the
etpr e
ai shed2t
Ph>if and B equae theproblem
expression of 4.4 (A2) theorem if Ph <1.
Let us set
11
mo.j
The
Te strict
iu-uunique solutionof
o theWD(F4,4E
eF)
which was constructed in theorem 5.2
can be expounded in the following form
(I)
0(X)
(x ')
Gd (X',X 3 )
p.rential Cd is given by
where Green
I i. X, >0 (Heaviside function)
3)
"
o(X 1 ,X3 ) at
I
G (X)-H
2 Ikh mo(sg(Xl)I')
X~
cTsIT
+
1("''3)
the function X2 vanishes at infinity.
X2(Xt,X)
tP
2 (X).
-(I)
Fh >I
chy h
2
sh2Y
G 3 (X) - H 3 (X) +
it
2- 2
d.
hkXo- Os t
32
k
shkX 3
k cos tchkk shkh'
W -cs~
-E(I)XV2
21NfKd,
Co
R
dX2
itd
chy(t)hch t hd
a followos.
o
-h dt
d-,
()
K
(K)II
where No and M I a,
~--
in1
/
dt+0(1)
i)
Rd"
two constants.
"
li
formula chow*
particular i.;.'I the
apjiit. ', A_(t) and i
-f couino and sine
waves at d(ow ream isinitysare
, by thy
ery simple followi,. relation :
A
r
ac~t) -..
t B(r
l ) y It) < 'r.u
where
q I,'
i5 a*~~
3'O
'
(ii)
tA.2
.g
Si
t.I
'(t
dk
112
r
ikr cos(t-)
o; the
lir4j -5The solution
hO,
) problem dej'inea in corolIary5.3
(s)Io ; "
(")l B %
Yt
chfIt IX )
.-
B(C)
3(x)
ks
0 h.H 3(X)--
yo2 E
00~
B,'(k)BO
.:)(t)
if Fh < I
-h
hkh
.'k
kh-
chyo(X 3 -h)
Chyoh
siyoX1
Ruo
B -o Yh
0
2I
B - 0 if
if X
ai
hd
fI
k2
(X)c
2y0 h
K
The above formula particularly shows that the
magnitude A. and AS of cosine and sine waves at
downstream infinity are given by the very simpie following relation :
c-is-i0
iA
AC-
]IXl
el
ki
M .l).( )(X)-iYoXdx
+'
XchkX
jllX
sinyX
1 )Y~c~-ifchyo(X
3 h)
0belonge to
I)i
(5.23) 11t;$ ,
u~(T_)I)
< C,11 l)
n
respect ;o data 10.Le': T be the solution operator (EM rs 4 ). The proof that is given fol-
0d-)l 1<s
(15).
(i ). This
2,
K
)o$
By. once again, applying Schauder's a priori
estimates one haa successively
(5.24) 11
'C
)
I
EM
d- )
11 to 10
25)
(5.
2, ct
Second step - 7':C ($K d-)
First of all the following is established
'C(
11(~I
) ';
Proposition .5.2Tc
&
(1)~_
2
md-1)
R
C (I'
8,
(5.26)1100;
mRd)))
m~aOcd-l
sauiu
o, %2.,
coepndn
V.P. (f,01 , 0,) prublem The followinR three functions are gives
f(x).x I,
) xcRd-I;
x2,,.s
1
11d-
2o cO~) IIchi
'VII
J2'a
.IIX
K
CI
)IIJ(VP2)
n"
2'
K
z (1)
n
(5.22)
114
n
~2(
)11
cptel
continuous.
competel
'
C,)11)
w(x) k,,
It
that:
, a.
xs I
?i sLch
t~''x_
ORd-I))
Then it follows
V
a-
3x
cOR
d-l~
c
Sixth step - T7
R*
'K'
Which is a consequence of the fifth step aod
the corollary 5.4.
Q.E.D. theorem 5.3
~O~- i
,1
-(mOJ')
mo
K
Which is a consequenc.- of (5.26)
)j
11<
V~"(l
sep-7
solution 0 .
Scaue pr
~.~
R,
~ ~
w, 7 it Is possible to obtain
Thetefor. ifu
from sequence 4.,a soubmoquen~o also noted~5o
asI
0which is covroti
M
h
ii
f
hc acrign
t~is sequence can only be
A0 (110)
R w
(P4)
Hetv, RWs is defined by ( ,10) where, to
ake the following easier, the not : ion change
X -(X' X 3 ) 's a - (a'.53) 'q us"d.
1
1
It is noted that VP(0,0,cS )- H~(h
0 )
he following theorem is a genaralixatlon of
5. J
Fh is .iatowd of any value if d-.3,
Theorem 6.)
Ioo
ifus
d-2. GNvo f~)oc
htfxPA;1XJ'.0 the VF(f,0.01 problem acwiif'I nf
qu~lto
JG fo
on
.a(
.Od-')'
aoX
;LR '(F
wn,
-re
'V
*
and "-M,
(f.
us t-
'nstants.
C "-.is c T I )
w dxpen.d continuous~y
on f, *1 ) i,.
,
he mapping (fti
' W
~)
iinallv .re eular)r-, ot the strict solulion it;pver, 11v ti- metnod use, Ll the
.
o lhp roof o f
to
'1
* t
, OR
,)
r: 1 x
):(]+!x'l
)-
Rivn t:nd
uh that -:
,x
looked for
-r
U,
IDP2) 4 (x)
2d-l
')
IXo2)-T-2-2,I f (x,fo
Ll--d-I))(DPI) 3)
(P3)
()
XD
C.
(I GD*
LIORd-1
unCl{Rd, ) is the spacy of messurable u(x') functiong defined on
for which the Lebesgue
integral
Flu;
3,)
LIORd-) l
is finite.
I I0
s -2
+,x,12)
I(*1x1)T
o ,,* (
C I(d-1
f(x',x)
zt-2a
C
2d-I
nqkii0a-:!gt
There
-ts
1
C mti:
an(, ,,tn
te
_
infe-, - ,-,,,
ire. jo,
he'
t ,rthe
-- ,nre
=
x)
Lenofz 6. I
dx'
jr'-
2 are Bausch
' spaces
5
+;
'002) C
(03)
0,(x')
1,
have
i( b 53)Ve
30 x
in(_
23
()
-,'ll,,
'
r (-i)"" o in0,T7%
1
).
-!'10 0 andK.-ao
f,
of
-f,, 0
f(x)dx - 0
(f'J
Aw
l
GId
o
D(4
(x
U '!' )
Whn 53*
3i,h- G
,3
-(
7
oaekr's, on IF
opace of dav
3aidd,
(
in a funot1on
,sa
asnate
-on
lrc,,aa,'rt
aORid
S
losi
t
eh
Mite in I
(ii) then the strict solutions of VP(Df,*l,2)
are found, i.. the solutica with suffi-
O(x'x
hR
e
(7)eh
)t,o
e
((x')
3)
mEelomfaotu
'pr,!sad
d-'. F,.
P(()p bt~ d
i
nique
C'(x',
Autim:
3)
The eolution
(i
8 Og'
.3 < h
(,
1
(f
.:-
aUoxV'XT,.
hence
a'f
c ',,
in
"
condition
dx'
r)
.1-I
wh,'
)3t,,
GD (X)
C
d-
'-
(a 'I)
L OR,)I
IOI)T
II,
d- 2d' .d-II
-t
,eua .2
p'(x)|
--
3.
T
(7.3) det p'(x) -
Ix
idea p.(x)12 ] t
h
er
c>0
cots
such that for o c < &
the mapping p is a homeomorphtsm from 1o onto
0
f1 of the class CI
;10D
m's(3o1
12 -n
' Rd be the mapping
(7.4)1dea p'(x)I?I- Eh
1101; %"(67o) 1
0h
(7.1)
4D(x)e 3 ,
x( 2n
I -1 h
axi.
h
wherevector
of Ox3 axis.
where 03 is the unit
t-
m"h"
'
Proposition 7.1 m' 2.
'
It
i as d t
t O),
There e.ists C > 0 such thatKftr 0_ <
the mapping p gas the fottoving properties
(i) p is a diffomoorphism
of the class (CC
(ii) x3 - 0
)L
3 ' 0
(iii) x 3 - h
X3 - h-
Proof of
%onto
2
cen
is proved with
r
110
)11
Q
lemm 7.1
Q.t.D. 1ecm 7.2
I)
(7.7)
S3
- f(X3 ,al)
(A)
p (11.)c
(b)
(c
Co'o(io1 d
p is of the class
12
p issabjeinfrom
a bijeetio r " 12eon
unto O r.
(c
i-
and
j
V
The (a) is obvious. The (b) results from the
form qf a
expression(l.l)of p which takes the
0
sum of 2 mappings of the class (ce' (1 )]Q. The
(c) is a consequence of the two followng
lema
[AMea ,1 <
r ;t'
119'?0ezista C' >Osuoh that for I
the differential A
ing p' isare invrt toa
Jf(X3*a
i aw
f.a7p~p~
" .
niu'fh
1of4d.0s
3O
70
"
D
( 3 -y 3 )# (a',*))*13(4
.Iph
s3
(a
y3
10
D(a ,'3 )- # ('
D,
5x3) -4(x 'y3)I
3)A
tf;t
If therefore r [0, c'[is chosen, with
h- 112h
Ii
()j
V'
(7.23) q
X-? 1,2
'-?33
kE
vanish.
all the others q
(7.26) q
(7.8)
f(x)
(7.27)
(7.28)
x - q(X) = X + E g(X)
'
(7.13)
d xd matrix.
p o q -X
The following formulae result
1.?
d2d
0()
731
ITX
(7.32)
ddfl
thtir tht~.J[It
s[l
J-
'
(7.34) d2(.
, dxd matrix
J
(7,h)
we obtain the mtrix equation
wo obtane te
n
9, ' '("t"
"
"
ladmatrix
a200
and
(Fht(0)) problem
f*t
c-T'
(P)
L( )O - 0
(P2)
in fQ
0
,-
Then it follows
.xd matrix
[~1
(7.20)
[.,.. ,d
(7.20
03
with
A-
(7.21) q 1 1
(7.22) q;3
xd matrix
k 0 3
for
d
dx d1
dX
(7.7)easily
(7.18)
ax)1
dx' dx
af(Ot(
with
d. )
17.33)
-
1 1Lf
q. LI
dd
d20 1 44
Then it follows
((..it
- [qk, q
Al @,
.Xp
(7.30)
Rd
P (Fh.Y ) problem
(
The problem NKd(Fh,c. i
1 t
33
3
alike formula for mn-t,2; all the others qkgs
vanish.
e3
cC'
5 (0)
('
W.
Jxl
(7 "
33
71
)t " ij
.2,3) or
aus
where~~~~~joiee
(1.3 )according to whether d-3 or 2,
I and 2(74)J,'
A - I for i
2,1)is
fr
(ij)0*l.2)and
(e)
(c)
~3i
j- 1,2
+
C'(9
c
2Cg 3S
1 2( 2
.+C
31
b1 (x)
(x)3(737)
i-1,2
cf'.( I-tf~ 1
3f;(l-f 3 )
&313
r[11 t
2)
32,
g )
-2
33j
range of T(c).RT(t)
i - 1,2
0
('i
-retriti0
(7.9)82
(7.39)~lase t
tf'x
+;W
with
X3.
' u2
index of T(t).
etc...
).id55.8 (rh,
,Z
iff the P (F
)prbiem
!A
lutO~ i of,%g'
(7.41)
ONa)
)M
$0
(7.42)
Tc
in particular
11
PF(Fhr( )I
F
ONKd(!h10,t.
VP(0.Oc -1r-
2
B ''~-
(7.6)
(ii
13-In the oe
I~a-I
),jplios
.*i3) Ut~
* 2 ()dA
The spae,%U.
of
(d.h)
(21
'ILI
IL : #I) - 0
so that
gt
(if)
ud~l
(7.45)flU0
~" ~~o ~
.)
600M~
UL
X,)
('1LO,
(7.43)
-Fr
(I)
I~i
-0 q(XW
is defined
74-Fraic
.
___1_6d___
1p W)
4.0
a uniqu: So
olai admtits
and we haoe
J
PoC Pw(
p Temai af pr ie
a
(1 Knd-l
areias fo- w 1
3
easM
a
7.
osi5o
0 such Jjt for 0 -t'C<
ere Wets E
probe
u2
;XI
-nul
th ,P(Fh.
ad
indT(c)
2)(-EfThe
2+
x)o~)
(W). E
of
.h
03spect
02
2LX
101
11independeat
oftE
of
rc212
11
1
g'33)nullity
'~
(7.38) B(E)
(7.40)
.c47IPIO_7.-
) 33 - E
) 3
(735ei
.CTo)e90
*(7.36)
ine.hn
operator T(c)istrcl
that
tii
fadS . I)
is a IsBN~c space.
/ii
72
T~O)- i*
of
eewdutue
the min/nsel
T(
T(aWit
nul T(o)
def lit)
def T(o)
Notations -
- I
(o)u
c T1 u +
E <
TI
T 2u
,.
to independent of u.
(o~o.
if
proven.
.o
Case (d.jh)
) with
-.
-c
t
in( .2
(2.1)
c)theorem 2 is
proof of theorem I
problem -
4f(O-dima'IL 0
2 T
If c o - ein (cote),
where it should beoremembered thai K is'the support of the bump, K is the interior of compactui K and =bdry K is te boundary of K on the
bottomi X 3 h :K
6K ubdryK.
The IP(*Fh,E,a,Ry) problem (Interior Problem) is associated with the NKd(Fh,t) problem
in accordance with the following
nul T(o)
def T(o)
Ind T(o)
nul f(t)
def T(c)
Ind T(e)
),
that
then tor .,
h
0. c 0 (the operator T (c) is not invortir1
bla except for sero element oZ Vil . Theorem I
Is thus proven,
Q.E.D. theor. I end 2
(Pt)
0. X 3 - h
02)
at " "
T3i
-I
e "J
e j
0e on F .
e
+
0,
and v
Lamia 8.1 .
nul (D-
2
0 in L (r)
.12
D
(8.)
(3
- --
(8.7)
0i 8 1
u.
Do'
-0by
definition of
(
t
is harmonic, vaui'hss on I and its
therefore
'. K thus
on Xg3h,
1
derivtive 2vanishes
i "D i
3 i slso satisfies an analogous relation to (9,2)
Space
_____________
H'ili) if
Green's formula)
2imit
is v;
f D in 2
The adjoint (opsrator)
xoci . n5
et
This is
is S
hdonth
of Stafin LIf)
and theadjnint
a consequence of ti argument expound in (Spill
formula M)
u -
(8.6)
*e 1 De ii satisfies
(8.4)
weakly singular
The ,perators '' and V are
2
a( )
operators from L2( ) into L (r),from
into 1L1 t(r).
= c
+
V
S
v
2)
at
admits
equation is
this solution
Ii its construction. This
unique
least one solution
lesauni
following
he
to
according
l2()
in
Therefore
-iit
(Mit IlI
-N2J
(D
D-, VIn
results and terminology
T. the following the
-f (34) will be used. We shall use,2the lact
that I'is a manifold of the class c ,O without
boundary but which connect op in a very regular
way with the bottom X3 -h so that Privalov's
theorem is true, see as an example (25,p.46)
for d-2 and (24 p.50) for d-3.
esofoerators
+D
2
hence by setting
L-it on r
S(8.3)
a- I thus
F2,1
N
(8.1) gives
(ii) a. y
30
(2
(8.2) 0
is
0 - 0,1[.I
Then it follows
18.8)
74
J
h2
0
0.6) itS
iqly rLst
(3.71
~~~~~~~~~~~~a
solatio of
r-zhe ?Ird iF[,,i
0
ai
t
p~t
;~t
kyF'tof
6. Also
is
prxnlem
rnyn
uni,-e nos Al ft
;-
There
3, i ii
reot tha
tIoe
0fsc!7
e-
th hoct
n
iemJ A.
R. !
Q~i0
th-.
31
iO
den.*sit
-t
V.bcnvr-oo;
Then !he tnttg%i.
in W"V7V
dtil.a
ul
hers 01 is A
Cd prob
V~i), l1,'
of tiso P1
I
b
In addlehg sr-nbc
.
nd 0.91 it
thai, (8.2)
(8.11)
3(X)
(3)
4
m-ete
;8l)
1
d ) 8..
ts the
ui-c
( (-F.02,j
pr-: 'en. Also, eqIli1t o
t
E., (8.- and (3.10, ::e obtained.
el
(Sit)
I
(()
-F)
(3.13)~~~r%
I1
lli
a pon
tion in LIMf
od
l
1.1
ei-.'sitinb
In nabiij I
it (olvs
7-iter..
woca in
!)
t.-aaTds a point o.0
'r- eden
unique solir
according io tnLofollomaing loxan:
equ.AtionlAnifts
the dis-
[K]rd
Ot
yoil-
integrni
.I
Q*o--nd-cordierg
(C*
.e
-A
limi3
-XY
fl'cc
1("1
(28.254)
We:
1(K,33
,8
i8.fl
ix)
of th itI
Lidentical
-he thr42;ee.'.;,e,
niqnez selution
ID
tJ
r-4
renqlart-
uveinotivLc
OJID. h.:ot-a
hiThic
f zher , 3 (i)
Ptotl
o'f
p)(+ph(C).1.2ffithi,21-2).
W.I)
Q1hl! t-ece 3n
tl
It
Kad te
s-nc -
This is 11
.ns.e.
icient ?rll:.ct
for thtnocia
a .I
0.
os
Lt
Acc-ording ito Im
z.
!thsintegral eqiuation (8(8! cdaiits a doilo:slIig
ton i
(V
Proof ofl emga h.(
in fact
tne
s5utisn
'
.4-
t,- (V.-
ia gull that t, 0
%caSates
hence, er-cording to the ptoportyol
rtaleP is th-. es, 2. 0,7-,in
isa
-14e]
',fin.
I)tl~II
(Sl)
is note "the-
,(nstant
lo
than
..
(oniqis"ess f.5
(2Vthere -t
it till-ra then
(Vl
1'-
Jlilesll
0Ic
Qimli
lena8.2
TS
ar.-sAwmuh
-
Al is in.
REFERENCES
(14)
HARBAND
J. - Three Dimensional Floover a
Submergod Ohject - Journal of Engineering
Mathematics. vol.1O, n'l, Jan. 1976
(18) KOTIK
(5)
(6)
BRAID B. - (6)
Le
in-Ke loin
La Problde
rtol~medo
ARD t. de Nu
Numan-Kevintions,
CoMptes-rendus do l'Acadamie des Sciencc
Paris. t.278 (14 jan.1974)
Lnear-and
Quasi-lnear Elliptic
EquaAcademic
Press, 1963.
P
(20) LAKB H. - Bydrodys ruics - Dover (1932)
(7)
BEARDB. - CtrVZl
nta our Is Probltw do
Newosmn-Eelvin - Coqptes-rendue 8e
'lAcad4siedes(28
Sciences.
t.276,
974)Contrjnv.Paris,
(28 jane. 1974)
cwun i.. ,mHLUE.D. - M,thod@ 0) Mte-
(e)
Phpsfoa
maltioall
Publishers,
19V.- eel.l,
, Intereci&=caISI,-Prb~e 2
(9)
MEN8J.C. - Existe.c.
Uniciti et Regula-
URAL'TSEVA,
LOSJL.
Limit** Xo fkcmogbiasa at Appliatiosno,
Vl. - Oumod, Paris, 1968
(2) NI
U,..u. - t1 s
,Ri~tana of
N.K.
(25)
(26)NECAS
J. Leo Mgthodea Direc.es en Thgorie des Equations ElZiptiqses Masson at Cie, Paris 1967
ABSTRACT
1. INTRODUCTION
- Us +*v
tillY
V *its.p. oh - 0I
121
78
bolundary cuondition
0oil
U! + o
2.3
-X-i-Z),itSp
"
I' s it o'r
V,1
ire
S,
V.
who,
.11)
i 'l.[ie1
orml
utardunt
ectr,
ieotwr iiinraiSc,
wher btounar co-. it)i
Iintin, isl tile tiotioio. Si. anidsile waits,
liS iidr
SA,. of atcanal are
('1
Oil=
it S
I jo
IVo 1 i
timl
1.71
5 -
P<
7)1
i+ S
1t1. 91
an
Siii,
1.1
ayi
teteeolc.i
prsuedlibloni
D,
SF1
+5, S"S
ll +.2+Sv,0
r3
SF0 rA
S2
II
SO
42
in D.,
U2 0.x
=
9on
2
loin
(D
for all )i. i
U a
"s
pg
x P. IX,
Si,
13.2)
on S o
n
Yo.7.) = 0
,)
S%%
ul
*,1xo.
T.1
F{
'atisly
1. 2. are solutions
V,
dv- "
+3.3)1
3.3)
0-ixx onS
rn~os
ds
Sl'uSP
D,,
iy
(3.8)
let
. and 0' denote the trial function%in the
subspacc of the solution spacein the snbdomains I), D,
and D,. respectively and let ,. T1 and ;2 denote the test
funclions in the subspace of the test spacein D0, D I . and
D-, respectively. We deline the funclionals F, Fl . and F,
in I),. DI . and ), respectively. as Iollows
V 0i
- 0j in) ds= 0
',
P iT, dst !2 -
+U
To
sP
ds
(3.91
Oinoi itiSIliU
tim
,,ds-
it',,
I'V ol
/
I
'r32
ds+ gi~
IIIj
'3
,2
he do"r iraar condition
and wi,
dli
J,
ondition
lim
02
x +oo
<
Tn dli.
=on
0on
Ft
(3.4)
Oin 0
02n= -02V On =
Oox
on J2
"
PF
(3,10)
(3.5)
-x
+ g J
F2 i,
.;
(7, j n - 2j,~ ds
i
W
(,D
dv.0.
(i - 0, 1.32)
, dR
(3.11)
+
Jff
x j,-, - j,-
g 42
2x P) ds
0.60
3
reduces to
Via
9l 0
dv -
D.
0,, dv - 0,
and,
i'
"
"I
lXe ITna.
at I IX,
-
. Z)
COTrrllitili
and
~~'n~i
~r~r
I)13
aItI', Isi,3
~~ril~
'i
nfor all
~i
(31.14)
0',1)
0.
'N'I
10)
F, 0,rs fr alitt1)
'
IX (Cos
kn
'1us
HEc
I eel ,
'in k0:
lint
0.
(37136)
1111all
bounded atsux
>~0,r
(I - 1)
. (18)
inlW2b)2.
ln= -
k 2
nil
p'~ +Air/2b1.
where pr
-ia
Il.
b)
In
,
ii) T1
(3.17)
. 2.
F,
ho
ic
tried Cialerkin equit ns (3.14) throutgh (3.16)
yo inti eicratioltil torn, in thecfollowing way. Let qIj. .
i. i = . 1 ix)
b[lie basisfor tile M-dimiensional sub'
J.,eof hlesolution spaceand let
gii
0.19)
ni
1 1
T2b 1 )III
fitit bnpH
tau
(3.20)
-,r ll~
3.1
falpi
[
If n > 1,. their is a' single solutionu for pi, for each value oif
Ul/gH. If n 0, there is one polsitive.A
'ution ((orpt~ if
2
U2/gH -, I, hut none if, u lgH> I. 'here are itfinitely
triat fosr
In thie numialr lprscedisre it iS itsiid
convenience tte yn'-plaut is sltift: lo oinceideevith J, artd
I-'
iad
F..
J2 for the corNtpotlon of
From the cotmplete setof vifinfuriuve
Ilic
basisfor
.i
es .Uli iytltei
)z).
7(X,.
3.2)
1i .. .l
14,~
~~~
ik
- si hF
x c~i t h)
1,,y
=)
U
+Iii
ciis
li cos
i y+ I /-
hII
+ III o% Lp,
(4.1
(is then given hy
.h)
i+
-kx s0
in
cost 11lis cos
-bh
p kil cnc e
up
2h (4.3)
?Im
7i
y+HIco n'rkxcItp
1.
t2
-2hb
x(.
sinknii
cosh p.(y +lH) cos
2+02+
y
4.1
0(.....0,,,'
aecoriling ais
=I
In(zb).lixp'
2
kin
Vr' cisip (y + H)tcos -j- tI- h;Ij
theiru
liy !ore
03.:51
.iivittliestagiiatioin point
ic
tIn tS
iS.
M
~
FOtES
~
~
~ PRSURS
~~ ~ ~ ~ ~ ~
4.
FOCSi
PRSURS
4. BLOCKAGE
MOMENTS, AND
PARAMETER
rsr
rew
v
IH
:oshItsnH
t
+j-j
rsr'CI, e-
nar
'
n tai
hnnH I
-
pH
+%in - jn
ih
a/
14.71
wthere
~*
-II~*~
for in>I
It ts of intement
to note that, fwr ateadyflow in a caital,
a -4
i,
IIIch
2h
-h
14.1)
n tn+
i
-
2b
os itnt I
cos
ny
k5
___
(I,eonkn.x+ ha..inknxt
eo onp(V * H) c
%a
2L
C, +
i4.tt
*)X, y. )
pie v tmidS
q
piaratmeter
U2
Intch in diacu-aed
by Newtiman
11976)t for the
Pressure
Distribution on t
U(
~~
I.
orl
>
it3are il Icigt
.tand Ival
tsnt,
& PoosleI Of 11t
tI
II
>
lcisiv ands p,
and
7g HNewinat
and1
IS 1,,,frI\I<
adI/I<
frIx
ri
(,l
~Witlet.,I
(1~~1
41151
FreeSairface
pW~
14.9)t
I
whtere
ato ile
act~ fit,, sittr tdeptiii
i11
a1,
15.-1
sipl lcetgit.
P IiH
resitectiset.
'P, Ii
Il i ) of ou
,sr comutatttionls, two, set, (ot Irile-ceetl
mtesht stttbsivossit, w~rc 05111 i rattttsl cljirsc 1, J stiii
un t
~i
tH-(Ittipn
H-
l
i d .1finei
1,0.1 notte.
t16is 7s t1 eleentts Ahtn the usv
id / is,
Inth
sets ofr n1iistisubdttivisionts. t1vittI'Tlruttia My tiii
illtiic
lions were ilien( 1 ItO e -ive
illr,sver, ik,
I lic
average CPU eseccotiouu ittlei for cli tFsIIiC
1i'Vtii (111111i
I esas Intstrumt~ent Adivanedi Scienillk
)I1L
thle
Naval ResearchIt taitratttrv was 11. ind kIsl,
for the coerse and tine meshtes, resptlecsustsedi
(lite costs we~re ;tttirotimawtey. St 0.111mid~ '10
resttectivciy. I
p1 it
it2 oit
v = g'1.
Oieeti 11001!
llW X. , al
11
iss reyitt
ittest,, with 1120 total eletieot andtt ill.
and11where p,
tilli resipc~i
Ill ilte
'2n 1.111
tle 1. Mir1toittttricat results Ire
cttttparetl
witht tile tanaltic reatit given bty 15.2f. As showns't
ilttattle 1. tijvit tinttc-&teoctt Ineshes wereuse foi fituitte,1
irtitsi ntiumbiers. ills I sltowu tilt illk acelirat sit tlie
ntttuericat icslill improvite% if finuer I-itttnitc tt'-it llttsltss ats
atti tii, titter mctsite shold Ke ilsetias (lieI rstj,-~
nttuer ttecreascs. I lite ctltttuari-tt
iti I-iptre 2 stit, pitso
atirevtttvnt ls-!wcvn tie taical~tetand~ iaytik resutltsitr
~It
;he Pressurr
Dirsoritulittsi
on life FreeSitri:te
lMIL - 1. HI - Wt, W/L 21
0106
Jmtc
0.0
0.975
009060
.
..
0.00926
ci 1UMERICAL
RESULTS
-BYNEWMAN A
POOLE (1962)
0
-.--
O.4
O3
0.6
0.7
U/./jR
(1.8
0.0
1.0
0.06110)
0.0071
0.900
0oem
0.0678
0.0612
Qow6
0.860
00475
.A
ooe
0.600
0.700
0.0830
0.0443
0800
0.017
0,0110
0.8001
Fignr 2 - Wavie
Reiitanee of a Pl01atut Daitribuitt
o
tB/I 1. H/I 0,11.W/L 21
Rintt
0.06
0.01370
0.0680
63
0 1677
0.080
0.0619
00671
0.066
00633
0.11
000D72
00354
.60
104b1007
Anelvtc Rtnnlt.
0.0612
0.0416
0.0009
0.0362
.)4
F
z
5.5)I
oil)
(5.6)
Ix\2lTable
Boundary
odtinnsndtin
726 Nodes 149. Nodes
Linear
ii
EnactHull
1496Nun..
-
0.O
600
0.570
03
0 500
0.475
0665
t.0250
321
3t
2.7764
2.260
t73
.60
.5
0.425
400
1,3847
oo0~x
0 11
*The nnndinnrional
a EXACT
0.80A
.22
562
3.200
2.7352
241
241
1 2240
0.7371
0.1791
1.4026
085
0.4
3.5
.15
2.8M9
2.3287
.84
.64
X/!Pil!
0.3
()
Kz\
L
can see fron. !5. 5) that tlie tine integral in 13.91 can bie
since it is of O).
Similarly one can also shuow
that tite rest of the integral terms in 13.9) should he retained
since titey ore of 0il 1i in tite nondimensional 'orns.
[Il
8 L2
K,
pH
,,/UH
It
0.790284
1 2.569032
0.790284
0,435529
-0.014165
!.745526
tilUH
n,/UH
bn/UH
0.736210
0.372829
0.728139
0.tt6152
0.017700
0.026130
4.417158
6.2880M
2.746611 -0.000608
8.163687
5 t1.043856
0.030042
0.000673 -0.064492
3.129618 -0.003859
-0.008078 0.007485
3.471880 0.310569 0.000784 -0.006729
6
7
11.825&V0 3.78285
13.10M6
4.07062S
6
9
15.491578 4.339341
17.578720)6 4.1092412
-0.0(1566
-0.O01t05
0.005677
-0.003588
0.000962 0000311
0.0007231-0.000285
0.067348
-0.000875
-00106914
-m0OD645
0.019063 -0.015158
0.008250 0.030843
-0.01894
-0.008766
-0.006122 1-0.002168
0.4
-0.2
NL
--
At
POTCENTERLIN
--
-05
-0.
CANAL SIDE0-WALL
41125
0.0
026
0.1
xlL
lmu 4~- 7b Velocify Futeallhb amnd
FreeSurfaceEilearu
l111L111
0.2. TIL -0. 1. HIL O.3. WIL I)
0.8
-L/2
--
L/2
F.P
-1.2
-r
~~0.53
--
-0.4-
-0,475
0.4150
-0.2
heuemeeni
pennre
Figure 6
cjalbiiin.h
th
W!Lall
fitareew
kn
on the bottom
lyth
-etrine
Iy -I
i
. 2ren
T L
.1c . tHIe
3
thjii Ioiltr heein
presure
ap~e~rndowstrem
uetjleti
ci thehulland nctis
notlium
,iowr Iinhshin
inbecase 1W ies~ir mte
be poit
smgci eficint
ni tre mns- ine4 5iiiiin thelrennrro
to/u -
016026
0.16223
u low6
Withi th linissuriad
BaaindaiyCondition an
-I0.425J
4.--aaii,1
0.0
0a.l1alBa6h749
d~n
0185itrutoi
the Fraud, numberstanted
the
i In a lelie
er
t nu~ cw ei in eenivs
n eprn
d
ormlas
m nethprevient PFUte
InfOiavninle ontal
flow ccua
oewtel.Iaslkerl
lleeeielm-b.aeue.I
m
Fan,-hnk
0.1
%howl
the
oicotunltui~thth
svn
pitir tIoietart"c aser d osremo
the hullsavnd in
laoe 4howninofhinrs
'ipnr
I t eauste
l
Ile jrof isid
the
tha theur
caact
computers lie., th
sar heralrnimnai
getutig largeins the pius
eecnton tie resetniffaiten menhd proiss a i gisith
msnan
irthile to lii
noicatedo
utent
ita reiemay
EF ER ENCES
~~~ai.
K.J. 1975.
"At uh:,tlt
5 t
rrrit'-/ herrernt3,i/r/
lwlth
Vadatirtra
f't t
bcmwiht/ttra.
:tlrtIrt.Nur
' Proc.
Sims[t.drod Nnim. Sbp Ioso b% liavd by~tttNavSdr
laewarcNavalhipvRirrtrr (enter. etesdaM, lii
lRati, KTt 197, -A
tSitir." sobted
q~r
rtttlk'
rflSip
J.
itrr
ae-rtlani j/rrrart
Jattar hutAlt, . 5r
Nttir Methd
in Iongat itter
ntom, raJoN97
titid
1. Shi Re. o.2 Na. 4uc,
Kr~irchni.Vori. I t, 1
ehauron.
We
153- 117cta
Japat.
afn
9.1ase.
-151
~t
aig'tajigtervXrtar.
Wil0.1..,Se
Wae an2Cane
Wat
hitt
,, Vrol1.
No/ ./tpr
Newm
n,
P ole
I.N..1141
I A P.,1% 2."Th Ii
Abstract
An efficient 3-Ijfinite difference
implicit scheme based on a fast direct matrix
solver has h-en developed to obtain transient
solutions of the flow about a ship translating
with uniform speed in a channel from its
abrupt start in calm water. Both "thin ship"
and exact body boundary conditions are considered. The problem is linearized in terms of
the free stream velocity. Abrupt changes in
the ship's uniform translational speed are
considered in addition to abrupt starts in
calm water. The present method can also handle
ship accelerations and decelerations. Higley
hulls, characterized by small beam and draftto-length ratios, have been used to represent
the ship. For Froude numbers based on ship
length . 've .5 it is found that the transient
approach tL ' locally .teady state about the
ship from an abrupt start is rapid. This
approach becomes significantly slower with
decreasing Froude number with the wave
resistance oscillating in time about a mean
value vhich is approximately the steady state
value of the appropriate comparison method.
as
LI
0-0
-T
'
n
2
-a
pU Ld
Figure 1. The Reference Frame
t= "Ox - n/Fr
+
x yy
at y=O exterior to
the ship
(1)
at y=O exterior to
the ship
(2)
zz = 0 in the region 0
exterior to the ship
(3)
at xO, Ll
(4)
at y=-h
(5)
at z=L
2
(6)
z= 0
0
On
n' *Ln
(14)
(14)
f dz J,tx' (2
h '
- W,
(8)
(9)
2g ,
w.
The initial
conditions representing the abrupt
start are
q-0; *-0
dz,
P_U_2 ,
at z-0 exterior
to (7)
the hip
h/ /l+h'+hy at thehull
z -zh(x,y)
(12)
2( f f -p'nxdS)/(pU Ld)
ship
2L
-L-J
(t x)(nx/n )dxdy
h'- Lh
and the limits LI, L2, and h of the computationalregion D (representing half the channel)
in Figure I are already nond ensionfl. In
Equation (10)the primes denote dimensional
LW
-
81
0
dz f (
0
-h
(15)
vAxx)dy
#')4'
SdW'/dt'
-
pUJL
(16)
KE
at
PU*L
F~Syd:
:,,-
+ff #nf~x-dxdy J
A
m+l
n
mrl,i.t(gl
(17)
. Om
=*j
z
d
d(PEidt
P UE
F-
Fr
S
2 dxdz]
KE + PE
(19)
(1)
+I)
2ii
+1
Y~infr(y-0oIsflowdb
will be specified.
'
y
one iteration and the cycles through Equaticns
(20). (21) and Laplace's equitlon are repeated,
using the latest vilues for #0+1
m~l
nmMl, and
(20)
(18)
Il.
m + At(ml+ F
ji
tf1+
FT'
mlJ+l
y '
90
.4. 1
41 .J+(
y
at y-0
(22)
point a Is zero.
ycondition
IV. Results
The results discussed in this section ware
obtained for the Wigley 1805A hull[Z) giwvn by
z - h(xy) - 3 (-256y2-6.4(x-xc)2
(24)
h
+ 9.6(x-x
Figure 2.
))
An Irregular Star
where I is the x-coordinate of the center of
the shi4. For all Froude number cases using
the standard grid, ship placement was also
standard with sc - 2.025. Whenother grids are
used, xc is specified. The Wigley 1BOSA hull
n
n
=
=
. nc + . f
2
U
g
for j=25,26,27
n =.7nc + .3nf
for j=22,23,24
.6rc + .4nf
for j'l9,20,21
for j=O,....18
(27)
for j=28,29,30
.Bnc + . nf
Fr W
(26)
g2
0.004
F,
0.002
CA 0.004462(ANALYTIC. STEADYSTATE)
t - 2.7)
CR 0.004551(NUMERICAL,
0.0
0.557
-_______________
STEADYSTATE)
0.004825 (ANALYTIC,
-CR
CA
0.-05002
(NUMERICCL, IT-O1
0.0
NLTC SED
TT
0.00.35
0.004Fr
0.W2
.0o
.0.32
00004261 (ANALYTIC,
STEADYSTATE)
1
PRSN NUEIA
METHO
rueNvbr
Fiue4
aeRssac essTiefrSlce
of~~~~
anltc
CR .-59
eo
aiu ANrLTId
ruenmeSATtE
STheD
tmsidctd
Cp*.047. Fr r*.57
h cluatdFrth
mllrFodenmer
3..05 n
rflsa
risaevAlusa lnsj2,16 n 2
4 h ueial
bandwv
1
weeC Cc46..070
nd.074
ie
fte stmnmmadmnrmo h
~~~~~~~~~
~ ~
CR.0C55
E) cusmsl
chage
'STEueThe
analyticIC
an0.0the02-
CR 00446 .0I0hs2
optton
eiaie
nlyi
taysoewaepoiedt
by th nue0alshme.o3r2d
CorsodnFigure 4.
theiswae proflues at thae si
feeigure
com1ut,
for
n
Veshow
s
Te fo eey10s Fode toutes
7
lern
ra
tr
o
ume
sh
thealredyiceas
4
applined
tov
thileaset~
wer
CR-.058
041,ad.0749 ie3
to
hatmnmmadmxmmo
h
SPRESENT
NUMERICAL
EE
SMETHOD
2
-- ANALYTIC,
STATE
STE.\DY
*
F,-C.557
-t2.7
-2
* obtained
-4 -
2 -1
/
Fr-nO 03
*t3.6
/
0-
\/
\/"tabulated:
Froude No.
-i/
-. OOF to .007
S.52
zU4
S
- tS
-respectively.
.2
2,8.
F.-n15o
s
0 t-21
'4 eFroude
I---
. ,..-
----
-.020 to .017
.557
-.021 to .018
--
.503
.4
-4
-1-5O
-.017 to .016
-~
.385
Fr-0.46
01-2,7
o
Range
........
Q/
One
0
2
4
am
F APART
Figure 5. Ccanarlson of Numerical and Analytic
Wave Profiles for Selected Froude Numbers
ia
ITERN
STATIONS I
94
h(x,y) "
O- (I-256ya)(!' (x-xc 2)
(PS)
0.0004''
-- - -
0.0002
X
(
0.557F
03
0
ic0.0004
0k,
0.0002F
0.5
"0.0004
F, - 0.45
0.0002 -
zE
z0.0002
Fr- 0.32
I
0
......
-The major difficulty in editing accurate
flow results at the hull surface from the field
solution when the hull is imbedded in a
Cartesian grid is that hull surface points are
generally not grid points. Thus although the
numerical scheme (which is subject to a Neumann
conditiun for # at the hull surface) Is secondorder accuratp at field grid points and is
efficient, the velocity potential # and surface
elevation n at the hull must still be
accurately determined.. These quantities are
used in computing Co, W, KE, and PE of
Equations (12), (16), (17), and (18).
respectively. They therefore can significantly
affect the computation of wave resistance and
energy conservation. Determining # and n at
the hull can be accomplished through finite
differencing of Equations (2), (3), and (8)
and/or through interpolation to desired orders
of accuracy depending on how many field grid
noints one is willing to uso. We have chosen
. w q w
um w
t:0.16
t=0.52
t 2.30
Wave profiles along the ship hullare c)mpared in Figure 18. The numerical wave
profiles were computed from the following
expression obtained by taking the outward
normal derivative of Equation (2) at the body.
,/
Inn
1/) 3.6
='
/Il
, / '',/
S'
"
!
~introduced
..
tJ),,,
'--
t5.00
(29)
Second-order differencing for nn using secondorder Lagrange interpolated free surface values
of q (with *nx)body known analytically) was
, ''
//
,'
-(Fr)
'nxlbody
,-
I'
1..
Lconservation
a
'
: ttnat
I 4l.11,
"
0<;
,<<
/: ,
'
(Fgu
re ,
and 21).on With
values
Ulotted
by20 coi~puter
the contour
Calcomp 936
Plotter
(ncremented
,Il
to.010
007
Fr..',32 and .503 were -.
body co,,dition
by the
'""".'-'';
',!:,,. ,..:.,patterns
to .022, produced
respectively,
Asexact
expected,
thewave
are similar to Patterns shown earlier in this
aper for the "thin ship" condition.
2.70
0=.60
t 1.20
Fig~ure 9.
lm
96
503
Goa
.557at t
2.7
0005f\~~YI
/
I,
-ANALYTIC STEADYSTATE
Cm 0 .004475(ANALYTIC,STEADYSTATE)
C, - u.ODM02IUNFILTEREDI -3 SI
S~o
E?2
CA
,-3.61
C,. 0 .000237(FILTERIED.,
-CURVE)
C5
CURVEI-L
o0045( - 2.7,ABRUPTSTATEFROMtREST
--
ABRUPTSTARTFROM LOCALLY
STEADYSTATEFOR F, - 0.003
-ol
is
2.0
IS
Is
2%
2'5
30o
35
40
Figure?
13. Wave Resistance Coefficient CR
Versus
___
I__________________I
06
ANALYTIC
STEADY
STATE
ut5n
FILTERED
FITER ED
.-
2.6
rlime for
Fr
--.
503
3.0
---
R. - 0,002616IANALYTIC. STEADYSTATE)
000OO4
Figure 14.
OM0F-.0
-A0UrSAR
AF
*ABRUPT START
FROM REST it - 2J)o7141
R 3IEENTFNLIltERICAL
METHOD0
ANALYTIC. STEADY STATE
Wave Resistance
i
7
.iiFr
9"
.4~~I
10
STAR"
PRESENTNIIA4OECAL.MEYNCO
IS
STATIONS441.6 FT APART
Wave,
Profiles
Figure 12. Nuaserical
for Fr *.557
0
now
2
IkM0
E O:
o -4
: :
i faSS
4t
STATIONS318 FT APART
STERN
t 2.4.
.RESENTIMBEDING
'
MTHADSAE
8D
E
0
P E0
'S
Selected~ INuMBs
"Tud
Ext
dyCdto)
BSRVE (FREE70
00006.AATE TR.M
I]
F:
20
,4
IIIC
t~l-2-
00
9.1ivS
0It
~*00Ul(1SgRVDP~h
002
V.
--
0~~
0000
TO
0000
Y1k
0. 000
4.
AIMIax0002
~00(w
mod2~
50
If
00050-
t=0.3
t=2.1
IG
4.8
\W
t=3C
/'
';
11v,
1
)jSqec3
o reSraeCotusfrF
20(EatBd5
FigureYE'
niin
t=48
AS
}101
t0.6
t= 2
t2.7
Figure 21.
50
V. Conclusions
References
1. Lunde, J.K.,"On the linearized Theory of
Wave Resistance for Displacement Ships in
Steady and Accelerated Motion," Trans.
SNARE, Vol.59, 1951,pp. 24-76;
discussion pp. 76-85.
Theory of Wave
2. Lunde, J.K.,"The Linearized
Resistance and its Application to ShipShaped Bodies in Motion on the Surface of
a Deep, Previously Undisturbed Fluid,"
Technical and Research Bulletin No. 1-18,
SNARE, 1957.
103
DISCUSSIONS
ofthree papers
Invited Discussion
E.O. Tuck
University of Adelaide
I would like to commence with a few words
of general congratulations o the authors presenting papers at this conference using direct
numerical computations, whether by field
methods (finite element or finite difference) or
by boundary-integral-equation methods. I have
found, on discussing ship hydrodynamic problems
with my mathematical colleagues in the area of
numerical analysis, a lack of appreciation of
the extreme difficulty faced by those attemptingdirect numerical solution for free-surface
problems. Professional numerical analysts tend
to be rather timid characters, who are everready to tellus what we cannot do for fear of
leaving ton large an error term, and who tend
to confine themselves to well-behaved and understood functions, difference schemes or kernels,
Our problems are beset with difficulties involved with the non-linearity and unknown shape
of the free surface, with rapid variobility and
singularity of the kernel functions, and with
the radiation condition at infinity. Many
herculean successful or almost successful offorts have been made, a number of which are
reported in thismeeting, including the papers
by Bai and by Ohring and Telste in this session.
The organizers have perhaps been a little
perverse in asking me. a known enemy of the
Neumann-Kelvin problem, to be the official discusser of three papers inwhich this approximation is used, at least in part. Before
returning to more complimentary discussion,
IEtme state once agsin my extreme view on this
topic.
There is no rational justification for
linearizing the free-surface condition for nonthin surface-piercing bodies. If the body is
hiuff. It makes big waves. If this is not the
case, then not only the free-surface condition,
but also the body boundary condition, should
be linearized for consistency.
Now an inconsistent problem Is not necessarily an incorrect one. The Neumann-Kelvin
problem must give results at least as good as
Michell's theory; it Tmt give better results,
In the spirit of pare scientific research there
knowledge in this area. I have the greatest respect for authors such as Dei who are able to
pursue deep mathematical analysis to a rigorous
existence conclusion, and for the other authors
in this session, whose tenacity in the numerical area is no less remarkable.
Discussion
by T. Francis Ogilvie
!f the advocates of the Kelvin-Neumann problem are successful and if the results are an
improvement over existing procedures, then we
might better spend our time in trying to justify the problem formulation than in trying to
dispute its validitv.
Discussion
WjTTi-ilweber
According to Dr. Tuck, the Neumain-Kelvin
problem is only of mathematical interest and of
little value in solving the exact irrotational
gravity-wave problem. in a recent paper by
Francis Noblesse, It Is shown, however, that the
N.-K. problem can serve as a good first approximation In an iteration procedure for solving
the nonlinear free-iurface problem. (This paper
has been submitted to the Journal of Ship Research.) He shows this by demonstrating that
the nonlinear terms of the free-surface boundary conditions are small relative to the linear
termr even for the observed bow wave, at low
Froude numbers.
105
6i
solution exists, this theorem displays the solution, but this is of little value since it is
expressible in terms of the elgenfunctions and
elgenvalues of the kernel, as an infinite series,
It has been shown, however, that numerical solutions can be obtained by means of an iteration
which can give useful approximations,
forrmula
even when an exact solution does not exist. An
example of this is the van Karman method of
flow
solving the irrotational, axisynanetric
about a body of revolution, which, it is known,
gives reasonably good solutions when a moderate
number of intervals is used, but yields poor
solutions when too many intervals, say greater
than 20, are employed.
Author's Repjy
by Sameiriing and John G. Telste
to discussion by E.O. Tuck
- -",ldlike to mention an imoortant point
that wa: no, emphasized in our papE;u
and therefore was overlooked in Professor Tuck's discussion pertaining to the computational time of
field methods. The high speed of computation
of our finite difference technique is due in
large part to the fact that only a very small
portion of the grid, mostly in the neirnborhood
of the ship and free surface, is "inve-ted".
Being direct (noniterative), the fast Laplace
solver used for the field equation permits us to
do this as shown in reference 4.
AuthorsRel
by Kwan JuneyBai
to discussion by E.O. Tuck
I would like to thank Prof. Tuck for his
comments. I would like to reply to his question
about matrix inversinn. In my paper, Gaussian
elimination is used in solving the matrix equation.
The number of operation-steps required for
solving a banded symmetric matrix is proportion
2
al to NM , where N is the total number of unknowns and M is the half bandwidth. As an
illustration, the fluid domain is subdivided
into 4-point quadrilateral elements. The total
number of nodes are I and J along the x- and
y-axes, respectively, so that the total number
of unknowns is N - IJ. The half brndwidth M is
approximately I (or J if J <I). The time to
invert
the matrix is then proportional to
2
2
N1 ! N . In three dimensions, where K is the
number of nodes along the z-axis, the half bandwidth becomes M JK4N2/3, where N - IJK, and
2
the time to invert is proportional to N(JK) 4
problems, JK is consiNI/.
In practical 2ship
3
derably less than N / , since a ship is much
longer along the x-axis thin along the y- or
z-axes. Furthermore, all the matrix elements
are zero except the nine non-zero elements in
two dimensions,. In other words, this banded matrix is still vary sparse. This fact can be
taken into consideration in the computer program
by using an 'if test' to reduce the computation
time substantially. Another feature in the
finita-elment method is that the time to comput* the matrix elements is much less than the
time to invert the matrix. Therefore, the nup-
i0
Abstract
The equation (1.1) expresses the gliding condition over the hull (M).
The equation (1.2) expresses that the free surface is a material surface.
The equation (1.3) is the ,ernoulli equation
in which the pressure is a constant over the
free surface.
The equation (1.4) is the non-radiation condition which expresses that the fluid stays still
at forward infinity.
The cquation (1.5) expresses tha' tt fluid does
not move at very large depth.
Figure I
The xOy plane is the horizontal plane which is
the water le-,elat forward infinity
The Ox axis, situated in the longitudinal plane
of symmetry, is directed towards the ship's bow.
The Oz axis is the upward vertical.
The free-surface equation is z z - h(x,y) - 0.
4 - *(x,y,z) is the absolute potential expressed
in the (O,xy z) coordinate systerc attached to
the ship.
The iee-surface conditions (1.21and (1.3)can
be replaced by the following comdition-, in
which V (a the absolute speed modulus
2
1
g
I. Introduction
C.n
an
-C.n
on th
(1.1)
2L-'
y ax
,
3zC3
Ix
h
a-h
in which
v2
(1.4)C
x
".5
4t 0
se
- -
(.5)
n figure
i).
107
-1
1x
.v
V2
Z-h
(
51.
A1
(4.1)
C.n
)r
*5h
+
0
f
4)I
(4.3)
--.
0
_
Ieo
as x
O as Z
O(M), the absolute potential in M(x,y,z) expressed in the coordinate system attached to the hull.
ix the speed of the ship.
C C
L
, an arbitrary reference length
A
, the outward normal to the hull surface
(Q)
X
- , the wave number associated with the
(4.5)
at
oiT.
speed C
nondimensionl wave number
L
C
_
''.
ce
(5)
X (O,k)
f(O,k)
number
function.
, the Froude
where
Is a fictitious viscous coefficient,
vwry small and essentially greater than zero.
h "
3'
a2
hull". Here, we only considered superficial distributions of sources and normal dipoles.
(4)
(4.4)
-
Then, in n ixing these operators, we will sasfy the gliding condition over the given hull;
this last problem is an exterior Neumann problem.
(4.2)
ax
Computing method
Kelvin source
108
M'I
=-
source of
which is the potential generated by a
intensity Q in an unbounded liquid.
-.
A
secM e3-
dO
if
(16)
with
-
+
IMM-I
4v
--M-1
jMMjj
(7)
+ iz-
k(z+z
do
(8)
idk
k-k O
'
'
[ec t
) d
0(I)
(li)
In
, if
1 (0t.i = EI(FI)
in the
In(t)
The figure (Ha) shows the free surface elevation corresponding to a point source, immerned at a depth 5l, oving -+ constuilt speed such
I .and having a - Itant intensity
as koH
Jr (12)
on,
t
d
(14)
1r
h + h
. re30
t i(t)l
2 a
2t
I
i l
9k 2
(IS)
2.
we have drawn
h&
(y._)2
where
clevatio,,
of the nearfield waves.
Q-
uthe
(r, ) b
g
being a function we call "modified en
ponential integral" which is related to the
classical exponential integral ly
n )
field waves.
T)
with
r
-f
(9)
jjT
R e
2-Qk
27,
11'
n-i
q x-3I,.)2,
2 + i (0)
asumi ngr
assum)ngh
= (X-x')Cs 0 + (y-y')sin 0.
and
$ (M)
Ie
if Y-y'
eif
(6)
'(M)
hf
y-y'
Figure
-~-3d
arid
-
--d6O
{) k
(20)
22d
kh-k
sec On i(0)
(21)
Co
(evi dipolelx
Let mx
rr,li
hnp-tffadpl
ihaigaconrstant.
intensity and a fixed direcseed
cnstnt
vin
anborzonal
in
tion
costat seed
tion rn
wth
s lErienta
clg wih
( -).
10)
T~iT
n
ez'-i
k
~iT ~we
iv
(.
an ,
or
-1i(a
I)r
W
HOM
ILei ul; repall tee following formulas corrampondlnq to tire potential due to a KelvinsBourte:
4, I10)
W-
X'+*
(I
2lr
nt
(23)
function.
1. (ochin:
1+
co
NP
41r
Sin n)l
Ml0'
Ic4-i(m CoSJ+m
(x
2H)O,r)
2 see
n--I
2.
)
(22)
PtB)
r~dM
1(M)
M
-M
-h(
4,
'sn"
'
2
atn a + 1 (0)
2
p'
ovector
110
q' ,r'
(x
-'
a PotentialI;
de
4-.-,kd
r
-,e .i (x cos q + y sin 9
(F'k k
2s -I
-,
k (x c
X e
s o
sin
tdk
and there.for
AA
0l,k)
The discontinuities of
substtituigq
Figure
or
nI
o+
=
'
e
kh 'i
H(0,k)~~ym,
1llO~h) -
_...
e e,+O~y_i l ,I]}
,,
3n
if
h[z-lisxen
(28)
12)1z
1 in the potential due
-,
associated with
1
I
) placed on a sue0") )i
(
iy'ti
Io r,sj-,t to t he
- 2q, for
= O;
froth
tensae
: k
+l{((,k)
A l(O~
).
291
y,,,inslij
The wave resistano, pferated by a sinilarity IlintriblUtio
is livn, by Hahvelock's formula:
-o 0 4 y
in,
n- ts
R,
in,
be modified anlnq
For the first
one, we
l|(mkb
vi'-
formulas;
(17), (1)
i,
the sal,,u of
s0.
dx dy
th.
~ ~~~~
that i i
H(6,k!
0",k)
00,k) and
(25)
k)
tran.sfors
202
of
and--- for
gentral i-ed
without care if the singularities are distribted over as open sirface (I cutting t hr plane
i
s
on:; ope O~Ot()I
a contour
c.
in fact, the theo-y
0 alo
i.eSenused to obtain tIle previous resuls Is
tiatly based On the Identity
itd
direct Fourier
z
0.
(I0)
(29)o.
or
2. Moditfied Kohinoul.
com 0 + y
dv
- i k (x
X .
- ccl))
til
(k
Se
k2
(27)
" 01 ,k) .
127) we,
1n
k (,[
- ilx
h
io
-L
11
il
H(O,k) -
thnt if
Let assnt'
a li-n
ditrib.lon
t
i-tr-hati's
e
a nv-c
ta:
r'
i,
plan'
: O, w
hvse
sIy
x s
In
in
11(0,k)
i,
(),l26)
-24)
one finds
in (23)
Substituting
I
i%
I, _ St
k , (Ik)
1(0,k)
are related to
and
= -
le
24/I
"0
iill
we can see
5)
(X)
Figure 4
The solution of the problem is sought in
the form;
0 1i(M)+
O(M)
(M)
3. Line
ntegral
(32)
with
is defined by
0=0
(M
Re
(Mi
2f
do
ReaI
SL
S
k z +i (
X e
+ y si n
)]
I iei
k dk
-ik (. con
+yY
ds
) dS
2, SF I
T9(M) being a regular potential in the whole
space z 00 which has to be determined so thit
1(M) satisfies the linearised free surface condition 4.3').
e
tis the value of Ion th, free urface (SL)
outside the hull
31 is the value of 1 on the free surface (SF)
inside the hull. (Fig.5)
Y
(5L)
wing condition ;
2
ax
21t
k2-- !ax
32r
z zO
2
1
immdiattly that
~Figure
2,
tf(01
W he
~2 $
iy
(h
SL
-c t'h
e so lu t on we a re loo k ing fo r i n
hrk)+e(
k(z+ll s 8
k-k
ax
+Y
34l
+ 1
-ax
a o
this
o -okN Cos
i
n
Wr +
(6,k)
2 oc
ysn(
((
ct.a 0
-{ik
sinnB)S
34I
d
e',
[a
1lrT
see0 . 1(0)
112
lk x
om0 + y sin 0)
"
we have:
Thus,
0.IOkI
2
2r"
6~) cn
.) 60k(6)
i
cs0(e
-)
-_
(36)
jfis
-'
'+
x
{ti
1 1
l -[t[21
(4-)
(41)
..1
20
-0
.. (xcosO+y sin B)
dy
X e
the contour (c) being described in the positive
direction,
From the defining formula (35), the modified
Kochin function is then expressed by
(k),
k 0 (0,hi + sec
X co
A3
~7
in
04
I I
) dy
The pntent-n( Of
... ..
the
~
(29)
l(6 ,k)
2'
integral, ie
XI
-ih-h
t (6,+)
(
ik
.oso
catXon
-6
I(
function'.
The terms
('t -
--
1)k(
k-ko
and(-
--
in
fo
,k~zc
generating the.
'10
.(40)
(1)
--
--
...
I-
...
iFk-k
+ i)
1(0)
-v 20t
6H01,k)
36'|
k dk
g
3
ino)
3
.
:a'< 0 dO
Rdy
k(z + in)
...
--c
e i2
1
Kw
) )IM.
j,)
dy
Ml
(f01
6)6,k)
0H(0,k)
with
OtiO,k) =
Fijur, 6
associated with the line
(37)
in a
1:1
k,
iw)2
. - ...
ii i ()
* in)
... . ..
.
k-ks~'O+
. .
k - k6 ose'i) 2,---1
d i0 ,*
-U :1
'
PIc
litta
F-ail ly,
oe deduce tie exrss
-ssuiiitvd with th' lint
(M)
1(W)
A(MMi
rM)i
dEW14)
1,'
eMi
(4
do
-_
IIm
In-,i
if tile 'n-reflc,l
'
over the hullI surfae
t
n r lt 'l . l
In
lout
III th-
aK lis m iii
M h ,n
iiir
tnil
(M',
IT ,nIIy
,,ia
Wi'tin
a iv
i -atiji isf
y (1
i~
hnitti
liti
,lint).
ij
",M
i-
-1i
,i
inil
tlt
hii
MM
Iin.i
...itIlni
I-I"',i
on l
i'(
~
,
i-Ii1 iiiIMf
n-al
(m,'I)
lipt
r iIn -
Wuf
ii is I o
I,
(nwn
-I
t)
tInith plai.'
'Sn
I n
'e'av
j'ttiitl
1 1
tc
i((
ii..
t -
rr,
- nt
t,-tt
114
LIo
uttnt
thp
iy
--
ints-
-n~liitt,ibuinriI
(, m M, i.n'
t' f ,
nli'l
Inn
t'~
i--t
'*
Ir 1
)-.-n
i,
&. ,irib
1n
iti -uir
.iil110(4nin
,~
irit
(41)I
0
nn14.-tr'
Ii~
I4
I,,I
of~
nrcI,,
ihi'nnlI
iy
llILIIy
ii-i.t
Ia1
ii-11
Knni
(M, M'.
i ,
(mtsf I
Idi
i
tinipI-1 intotity
I,
1MM
nce).oers,'i
liii
f-,ii ( )
ori sr
hal,
rit
,nnr
otrr
-
to
11
,ItnIru
ornu)a
I1,h.n
otIn
dins
KIlvi
- ni
fjli-n
I,,
I'tI
liiiri
IMM'idl-M'
.?fI
() mlpr
trihutilon
fron (18)
pol
-(
EiA')hMM'C.dCtM)
(4'riux
of
.
.''C')nd
isprfca
ittutoso
or
cepaotential.is:hsedestis
raea
(50)C
im'CM.
N'o
where
ACMM'
(MCM'h
fthtto
oefise
=iieThe
sorc .
1(M
(.
GrCM)
s osensi
swn
sie,
from
threor
whr
(
,m'
th
alrad
(75
w'fno
i-
to stil
Gree
to(I
~~
Mere
corditit, qilin
ine directl
presse
thatis theown
p veia
rateu
hulltv iseultoteladn
ME
eachMsin I
C of, th
hull
usnwom
inre
insit
fnte,
i
e=~C
ex-tlbl
qette-ll
al
Whe th
nse
qua
by
wCa,1
.C
am, t Co
. 1
oxl
tplsst'lst
iin
It,
~r
1s-rt"'l
cA r
nt(r
t searmsMsTxW
Cx
ttuqidntell.
ex-al
r- ri
tothele
'
and er
writ
MM
rcfential
)C
sdircl
t!,C
'
to" torte ti
setn
thIs
tltt
inarto''o
hatI'sttent',
ha
orhe
~inside therlWr-
4. EM)uIxa
rata
th hllis
isid
ti whc"
(C. W5)o
Cf)ai
Heto
eurs
y is qnowecul
wnmoagt
i- ttn( m
tittt. (m r
mf
dettatie
tetei
rtt
L)i
N5.
"7
0 Kn
Ni +
X
I
~1
'l
(CJ=M
cients
-H
1,2... N
1
Ki
(M , M').dZ(l')
1))
160)
in order to
fonm numerically only the quadratu,es which are absolutely necessary.
(61)
(
x(M i , M, .de(Ml)
X1, =
le polyqona'
The coefficients Kij are called "influence
coefficient"(f the element s, upoc the point Mi;
lihewise, Xj is the Influence c, fficint of
tie arc t1 upon the point M i.
In the case of a superficial distribution of
normal dipoles, an additional difficulty appears
ti,c te fact thit, in the licie integral,
,h(,
ie terms depending not only on the drnstty, u{M'
and
elements.
al,
)compt.ted
.)M'
o-M
(M.M').,I'lt(
N'
which Is II equations (SS) an;) (5f,), th, diff,culty is ilnmmdiatly removed iii intecratin
by parts ; As o is a .n)form funct i)n, at Ve.t...
tot a symmetrical hull n...inq without inrld...c,
we simply 1cn-
K
13
k
ij
+ 'k
il(2
(62)
with
(
'
-,
hk 4'1
is h."
cttribution
unlie
2, 2
of th
cut only at
the cent,
H,2
r1
11
h)
i'peoah*q,9in
trm
*1
so easy" to co'mpute.
X se,40
A way which cculd be usee. to solve the diacrtised prohtlom is to cone-dor the tensui(J
for
In ,
I
.j " M -, Additional uniknowns t at that ,sent, tile condition
- Cm
a or
- 0
hould be
satisfied
iI
) dFiM')
8'
c(M,M').dtiM')
jj-.j)
jA
ti
j.
1i
;,w
di(M'
()6
i"'t
of theI element
kith a unt.
lie
ittensity.
cos28 0
uid
Ls8ad
Jo
(65)
c)
(71)
cos6)dy
- 0
The contour C is then made of m segments 31,
k
whose ends P1 ,k
P k+ are numbered from I to
m, turning in the positive direction ; I is
then
20
(
con2
(72)
)
)
-ir
(p
or
Sp(xj-x o
(70)
[{g
dx
G()
C
2(r
CC(C) dE(M')
i(e)
2
2
g+Smi 2 cosfr
+ qj (yJ-y
1
--z)
=0
(66)
r.,
is the unit
ko
1j,k
with
On s7 ,
C =.
the variable
, :
(xih+i
z -zj+i[(x -x
sec
)cos 0
y1 -y 1
- irsin 0)
(v-yo) +
# 0 ; from (66)
y,k+]
(
yj,k
(
On a straight eiement, 4 is a linear function
of one of the variables xj , yj or zr; we have,
pj
integral
I can be writ-
1j
l
dxs
d
d
k+i
I
co s
p i+
~(,k+I
I'>;---i
ly
1
,
j
We can use
c
o
I
kI
r'J,k
1l
dr
J,k
and therefor
k
j sin O)(sJk l -
(p
or
2
cIs
p8 - ir
d.)
.
j,k
(67)
rose0
(
(69)G
-__-___P____s_
,k+(
being the contour of tle element s, descrid in the positive direction detormie d by th.
normal vector nj
C
way
117
k+1 -Y 1 ,k)
0(,
(74)
For CJ,k+l
il~
jk
Ir, - Sine)
8,
5
.k
-ir~sine~x -xJ,jk
k)
-YJ
k)j
Yk
Cos08) (
(pj - ir
Finalily,
We hae
(q i-
cient k
0,
equation
N (8) -
(75)
G(
(21J.
we have
et
ek 0 sec 2 8(2-
H1 a)=-d
(h ous 8
Y sine))d
1)
IM
ko
" ( ) dr
2j
iJ
Ki
i(-r
sinex
8) (x
ir
{x
(a) sec 0 dO
Kis
cn
8)(
j kl
-,
ij
__ J ,k+l -Jk
k)
C1J 1
G(C.)1
lC .k j
i
+yj sin
From (71)
we deduce
(76)
Hj(8) ff-
(6
[.
if
scs
k . se e0 [Z ,- k(x , Cos e
(71)w e
C
m
S(p-ir j
~ [e
G(0
iros0
r,
( pi -ir
in
sin a
1 ou s
dx
d5
with m
sides
Kijk
L(J
)G(
J~k
~26
Hj()-
4
j.k+l
8)
)(qj
and,
.Ir
k-
is
n )(x
+j,k']
bt
kJ
J
f
JJ
is
0
a nmerial
to
if
-4
j.k+l-Cjk>
It
6.
o mutaton th
v~ess
tanc e
6. co
After deteriini.g
ki~nmtc
resistance
the sin"ularity
ly equivalent to the bull,distribution
the wave
is calculated using Havelock's
for-
sec
3C 0
JI(0,k
ae0 eI
d.
k0
In
a2et*I
y
auM.rqd.cI
tts8
!H
o',upletely
I77
Jk
-.
obtain
in which V (O ktec6)
U (0) ia the value
o
taken by the "ModifLed
Kochin flinction"
for
k
2
k I 9ec 9, TO start
With, we write the vals
(78)ItI
o(78)
]
it jj.k~l-Kj.kj
2 o15
j ' k
[j"
.CJ k+l+.4
tant distribtion
~s2l(e
Hi)
of eorele
dipoles
m
J.;[ainO(s
w k.
the contribution of
- Cuaosa
-xk
-.
1
RJk)k
where HJ, k is
the function
defined in
equa-
tion (78).
Finally,
for a segment
contour cintersection of
ne z -0.
we lace frame
belonging to the
4F(8)
(39).
140) and
(41)
S(e) -7-
op: (yj~
) H
(90
1(80)
341
pj =
i
Figure 7
with:
+11
(e
for
"J+ + eC)
H
j
-Cj+
T
2X
-
ik asece
+ 0.0
Y2
-0.2
+ O1
Y2
x0.01
-o0
for
+yX sn-
cosa
0.01
-0
Y42 -
dicular
(8)
sec
-y
- I
Y4
0.01
is
Yi . O
)H
= - 0.02
Z 3 = - 0O
0.2
+ For a superficial distribution of normal dipoles of unit density, when the hull is perpento the plane z - 0
--0.2
z2
Zi =
being K,
0.0)5
25.
+4w k
pj
4 P1
(a
1 -x
C
e
study
is still
IV.
of care; when w
1(82)
very important
very disappointing
to spend a lot
in establipeculiar difficulty
is no
The
shing the computer program solving the Neumann
Kelvin problem for a singularity distribution
made of sources, normal dipoles or mixed, Mowever, the computing times are extremely long, due
to the fact that each influence coefficient is
given by an integral whose kernel F()
is a
very oscillatory function of the variable 8;
when the influencing emement sj and the influencod point Ni are situatd
near the free surface
the high frequency
sacillatlone occur when the
kernel values an
still
of imortance. On figum (7), we show the kernel variations we had
to consider In a concrete example.
119
Figure 8
. The computation of the influence coefficients has not been performed with enouoh accuracy in every "ases; it is not totally excluded that peculiar difficulties could escape for
certain relative dispositions of the influencing element n9
1ad the influeuned point Mli.
Now, it is sufficient to render the solution
completely aberrant if only one influence cootficient is not very well computed.
Table I
-.
Program
60
888
7151
144
920
781
-
"
192
-
(8).
Prog
894
Program
S +D
708
783
-
782
-- -- 786
---
It appears that the results are nearly idestical when the problem is solved with a dipole
distribution or with a mixed distribution
however, the wave resistance value obtained
with a source distribution is very different
from the preceding one.
Besides, the values of R,, given by the program (S) and the program (D), are very rapidly
varying with N.
120
the
Table III
i j ;
sis.
-------------------------t
mean
924
984
144
60
obtained always
the table (I)
the last colunn
Ranalytim. 5
al
Cal
1
192
288
540
982
986
82
computation per-
formed by C. Fareil.
b) For a singularity dietrbution of a given
type, the gap, iR w , between the values computed
at exict Froude number and in the hypothesis of
zero Froude number, seems to be little sans
e to the hull discretisation (Table IV)
Table 1I
(zero Froude number hypothesis)
Program
S
Program
D
(028
320
1037
9ARw
60
Table
Rw Analytical
_______
8<
143
0C
1100
0
io0
80
Theory
c. Farell
_O
900
- 14I
140
______
(sources)
the interval
IV
/N6
I .-
Program
(S+D)
Sources
o sources
same ac-
M Doublets
7U
0 Doublets
i
1oo
exact
e:o
exact
zro
N
I
200
300
400
500
Figure 9
Co
the half sumnof the values obtained with a sourc distrlbution and a dirole distribution is not
144
(Table 1I).
121
:I
13
, r - I meter
H - 1.6 meter.
3
Cw omental
a
y(i)-r
p
w
-
(
*
e
t
,2 exact
.Rw
exact
zro
Y zro
O),)V
D(D),
0
0
13
I
0,35
0,375
0,40
Figure 10
These results, given by an inexpensive computation, show that theoretical values, though
they are scattered according to the program (S)
or (D) used, are not extremely far fiom xperimental values, as they should be in computing
them with the zero Frcude number hypothesis.
V. Conclusions
This study on completely submerged hulls,
even if it seems to be far from our initial goal
gives us some lessons i it shows that
a) the difficulties, me' in the case of a
strface ship, are not totally due to the junction between the hull and the free surface.
Acknowledgements
We were applying ourself to the writing of
this paper when we have heard that Admiral Brard
"Profeaseur Associ&" at the "Ecole Nationale
Supdrieure de ecanique" in Nantes, was dead.
Our emotion and our sadness, due to our intima-
te relationship
122
References
BPRD
[121,KUK Y,
"On the hydrodynamical singularities for
surface ships with special reference to line Integral."
Doctoral Thesis, Department of Naval ArchiThe Ui.of Tokyo 1974
N.tecture,
[131 MORI K.
"On the singularity distribution representing surface ships"
Memoirs of the Faculty of Eg. Hiroshima
University Vol. 5 n' 2, November 1974,
p. 15-26.
[3] BRANDN.
"The representation of a given ship form by
singularity distributions when the boundary
condition on the free surface is linearized"
J. Ship. Res. 16 (1972) 79-92
*
[14) IAEHAUSEN
.3.V.
'rhe wave resistance of ships'
Advances In applied mechanics. Vol. 13
RADN
p. 93-245
[51 FARRLLC.
"On the wave resistance of a submerged
epherold"
J. Ship Res. 17 (1973) 1-11
[6)
FARELLC. - GOVEN 0.
"On the experimental determination of the
resistance components of a submerged aphecold"
3. Ship. Res. 17 (1973) 72-79
NSS J.L.
SMITH A.N.O.
"Calculation of non-lifting potential flow
about arbitrary three dimensional bodies"
Douglas Aircraft Lbmpany Report E.S. 40622
March 1962
[11) KOfI
r
. - MORNGAN
R.
"The uniqueness problem for wave resistance
calculated from singularity distributions
which are exact at zero Freude number"
3. Ship. Res. 13 (1969) 61-68
123
-Academic
COMPUTATIONS OF THREE-DIMENSIONAL
SHIP-MOTIONS WITH FORWARD SPEED
Ming-Shun Chang
David W. Taylor Naval Ship Research and Development Center
Bethesda, Maryland 20084
ABSTRACT
free-surlace condition can be taken into account by m+odiiyingthe fundamental singularity. In comparison to the zerospeedcase.this modification only leads to an additional linesingularity distribution at the intersection of the ship and
tie free saurlace.Numerical evaluation of the modified
fundamental singularity k similar to thz computation: of
+x4
and
*+
x-
.1 U
+g
an
0 at z
at
wheic
0.
g is the gravitational
acceleration
(nI ,n2, n3 ) = N are the components of
the normal vector N
=
(n 4' as ,n6 )
(in, in.
FORMULATION
and
r xN
m 3) = + T V(X + 0)
rxmn - V(x +
FT
ff
-J P dS
Si
atOT-
(9a)2
T
tl)
(2)
Celnt,
0
o -0
90.0
in the fluid,
dlF(6,k)
n
dO
dkF(0, k) +
J L
do0 dkF(o
JL
bile
k),
(
0 1
con ibly - Yo) sic 0)
ke
F(s, k)-
gk - (w + k U cos 0)2
(3)
(4t
(8),
if0 <
=i
arcos -
dO
Rt
where
6
0 and-t
4
j
i/a
0= UW/g,
1
and
J
G(x, y, z;X',0, z.
7.) =
withLaplace equations
2
3)
with
P/p
(6)
if0
i.
ini'l + Unitti)on S ,
an
k2
k1
Li
k3
20 cos tiS
'g /g-
+ v/
0<014
0
I/g2-' I - -4cos
, cos 0
2
2
R = (X - Xo) + (y - y)
2
i = (X-o)
+ (y_ yo)
+ (Z - Zo)
+(.+Zo)2,
and
(x,, yi, zo) is the position of a singularity.
With this fundamental singularity, from Green's theorem, and
the fact that the potential 0 satisfies a radiation condition
at infinity and the free-surface condition (6) qt z = 0, it
follows that the potential 0 at any point p inside the fluid
2
can be expressed as
0(p) = j
IG(p, q)
--
(q) - 0(q) 1
Gp, ql dS0>1/4
I
mFigure
Jf 2icUG(,. ,) 0(q) - UG(p, q)
2
+ U 0(q)
-1
ax
sq)
G(p,
q)j dy
p
(9
FfG(p,
q) Q(q) dS
4w S
0(p).-
!-
G(p, q)Qlqlna
9I f
qldy
t101
0
jo
P/
Y] I
t -
[UVlx
0n)
tc
V(l ed '
2
where the last term on the right-hand aide of equation (t)
is time-independent and aaaociated with the wave resistance
and the lift. The other terms in equation (I ) are time
dependent due to multiplying by gIwt and give riae to
unsteady forces. Now, let us denote the amplitudes of
these unsteady forces aNd moment y
U[V(x +
PJ
[Iiwo+
0)
(12)
dS
and
=
ISix'l
jS
'S.
(13)
NUMERICAL RESULTE
=
(
o sn
on
one has
F
"
pr
p
t
S,
+ tU[V,,(x + Qn
V]} "NdS
((4)
and
M pf
fiw
-a
(1)
A. WIaN Resisane
The wave resistance
ofa submerged body hasbeen
evaluated by Chang and Piea from a doublet-distribution
3
method. The present program with wU/g = 10, and
2
p1a/g- 10-6 .g/U was applied to the calcu!ation of the
No)]
dS
UO dRx V(x + 0)
(16,
and
am p
r{ e
.Froude
SB
+ U0 V
0 )xN
PJU r x IVix + o) x d 1.
(17)
s ah
b
ari
-rnt
d VugtSo1' fosreries
b -on0u&ip
by CGerrltsma"
and
Serieshv
60, (i
II .70+ship
models. Experimental data as well as srip-teory
predictions
for this ship model are given by both Gerrittsma and Vugts.
It is known that, In the practical frequency region, strip
yields
gd motion predlitlons fur this classof
shipsl Hence, the present programis applied to the same
ship for verification. Figure 3 presents the added-mass and
damping coefficients. The strit-theory results showr, In
Figure 5 were taken from reference 17 and thepresent
calculations were obtained by representing the hall'ship by
49 panels.
The solid lines in Figure 3 denote the strip theory
w +
)0
on
(l8)
the results
of the present calculation and the measurements
127
1.4
1
1FRICTIONAL
@ FORNORMALDISPIT & TRIM
~~~
1 RSDAYFROEcTO
~~~
&TRIM
0M
PRSN
It
MEO
-
RM
EDD'0
41 0.8IUR@FRMDLFRET
0O
I
1.0
L IXE
00
HI TER Y
0.8 0TI
I
0.2 RFi2
' r
o$rio
00.1 -
ofMasrdan
acuae
lio
Wun RTtlac fo WHle Hul21
o f/d g 1e.rmet,0
setina nd0s
Moio cofi/et aepetwc
aslrestemeuD2vle
h srpter
forIh
,e ndoneobaiedfrm
wol
hesu o
th dmpngcoffcintfrm
7ri
teoy
ppoPma2l
e y wmde.
l ih
ns e band
ZwA
rdce 7talfelecisicu~gte
aiu
rqectepeitdvleo
yaw,
an o respectively;
apnesrmnsaeson
U iefceiso ichtr usa-ul
wl h
rrunrs
h
i ni r
la
un
wy.Ilsenled ee rmFiue3thItr
peetcaclroldmpn
iefirn
peitd
ysrp
hir.de
i
oraddmseofplhhev.iThev
M
n
reit
iesa o wynhdrtir
uliinsvr
re
i h t lredre
datamotTHhove r~ec
0.
e IN rsntd
S IP
T4 e. or
th0 reer
rri aittoimto.A
em a
vaue
of0fo
of e'.
Iusi i otatt
h
u e n o itenrceuce rudb
ietire
rsialbir
s a
sti~hoy rdcirs hc
eiaecnieabyfo
he
dr oepeierldifclis
el a
ie p e
measurenPRESEN
f0 h ic-raemre MEcept
o te
i tr i s
o
o h rdcino
nufcec u tirter
orde
ofmto0tfeunisaoeSsc.srpteiylr
~ ~
~ ~~ ~ ~ ~~~PFE
~ ~ ~
dapn,
h
n retecisiswltk/w.I
roici crfucetsi
imao
o esrwnhwwltreti0.r6nsrnlcacia
treicsfury rodmtoncefiiet.
nte useo
rsetclultosofpthage
el ih
th
prireta
e
dta Hwe,4
fu ihrmuts rrt
swyardsws-ril
teprdciosnr utvr grst hy
devite
theneasredvalus;
rirm
igre cl2
approimatly
tentypercet toer tan
ahw:oeotie
ate
ntrf
rept ictwmeasurements
beteethe oeuude.dl andcalcul aned
o otidcniitrretepeitinai
h Ipovmns r.
h
t valuesareso of Mea
eat Resisndc
for
ie
frmi
surend Cacltedrytistatt
r
H thele
l hreiessl
aredreemente
frequency, t h
hellswre
dampings
f
Ch
o
rirar
peet acuairn ndcteta
ee
c2892ir
aiu
he
rdce
oeffiqcit fo m si t
iIri
,tr
hiudb ibius
o
Fm aier proxiae
ly
h wyad wy
dmigcefiinvr
alreue etrrfs
aidhnmowertfre
h
a Seiea The
avntsway It is seen
httremmeasured rethats tie
from
thenwoe mdel.
mas andre e sred rlhte ri
eta
frtovter mst of the setons, 0 reiotrseery difeen, It
seemes
thatfr theaaue6 roll d
imin
co nta may t
besutip-eny preicilorwhc verigate
edlctlviiusymromth
measurements arep rer te fitcer
eati o.
pitherd7
Foe
mdespof moeftinata reduences boe5 strip
e
theory
are notfiiec
ser
grodcthefal-oy
redition shefifiets.Itire
ae of
incul-olldmping
wfiecet
pereescandby stri
degr ares howt
agre 4.t tige 4eauetr
ashi
w ela
the
slt
ditccltions o
usngstr
theottry adiesoa
pveultiamtoa. meAst
metse d pblcreioeuenyionc ointhigurec4eprecthe comedas
ued in Fxgure . adifwiltoie api
lae oFiue
ti
enfo
h fgr httepeitrn
strip theory
the prevetiomuttiona
fnd mofrt
agre wti woficth t leoa
frequencies
l iswllkond
all
maexeim
mode
lion
an
iepe
dapin
a
f
stemagnitude
of
rll,
thee
amping
d&Cltdc
coefficient
camt~ns
excproteei ya.
ando
flow minoet,
thel moob
e ir
fom,*hm
ul
I.rxnieytet ecn
oe
hn
horsodn
edfrtetrediemnlc
eovos
decse ro temesuedvavi
hec~cu-dvaus re
THEORY
-STRIP
E 10
PRESENT CALCULATIONS
Is
PITCI,
50 0
~40
\\""
30
PITCH
22
0
0.3
10
~*25
0.25 0.2
o20
15
ROLL
HEAVE
00
Q 10
05005
o~g
0
01234567
01
3.
150
2.0
25~
0.
00 A
12
100-'
60
FRQENY
Figur YA-8eoSedMto
ofiinsfrSre 0
120
8Yw~'
SWAY
, 1
.0Hl
oe
2.0 --
1
SAY
i -
'
1.5
.5
, 75$WAY
SWAY
AtM!4
1.0 -_
1.0
L.o
O.7
--
. 1
0- - i
0-01
00
OL
00
1.00
*Y0.25
'' f - -
-/ ROL
-0.26*
z0A
0Ir
0
R:05 -
7- 01 0
i.C i
No.-Ob
IR
00,
YA015
.2
~0
-t
0,
,I
YA
~0
0
0
01
Im
II
by Waw
4inliud.
.0
0h
10
, Iaw
.I
lIlIO
flbshmm
w.
STRIP THEORY
o MEASURED (WHOLE MODEL)
0
PRESENT CALCULATIONS
.
I-
ITCH
N 10
40 PITCH
S30
o4
13
-20
O.
0.6
010-
25 -~
0.26 -ROLL
-0.20
20
HEAVE
.15
415 -
00
000 00
0 10401
0 5
00.06
00
PTCM*HEAVE
2,
WYRL
2..0 -
CO
0
220
0*
4S0
0
0
4.4
0
34
01
I -rI
_ T _ _I
(7
-
E 5-YAW
5
I
100 -SWAY
60
15 -
40 -00.
010
Oo
020
oC0
012345
012
45
FREQUENCYwlsec*'
-c---
Fr =
Fr = 0.20
12
E 10
8
"0
6
2
a
0
,
PITCH
0
I
30
-E25
I
x
20
a
15
15
0 10
w
HEAVE
I
I
0
0
2 3 4
5
FREQUENCY,
132
THEORY
-STRIP
0
*
121
1 1
10 PTCH
40 -
PITCH0
20C
10
-0
c25
0.25 -ROLL
e20
a
~0.20
HEAVE
~15
.05
0067
~.
10
00
O.
500
0
12
0 0
3 4
0o
01
5678
2.5 -
MITCH449AVE
S2
45
23
WAY-ROLL
0
2.0 0
*0
105
0-2o
00
0
00
00
.4
0.6
K-YAW
~20
-100
,I I
-SWAY
sao
10-
I Y
0 -
-o
zs
0
00
40 -0
0.
01
0
23
4 56
FREQUENCY, .(sac-)
01
56
2 34
1
FREQUENCY, wisec )
the speed
effect incruass the damping coefficients. The
vau~ofth
ceflistnan
amin
elw ha feuecy
Th
omarso
F[
{i2xu
L\L/
I[lift.
(I'll
0.7
- 0.3
II
|[Ii
II.
.
...
-a-J
-
R Nmuch
PRESENT
CALCULATION
NEWMAN
MEA u,,~tC
M
ir
rise
z
eero
forward speed calct,tri.ns. that, for most of the
freqserrey range, theagreement between the added mtiass
coefficients computed by the uresent method and the
experental data is better Iran foi, strip theory. In general,
ii iislts irrtictcd hy the present method and by strip
!///'
0,03
Sior
F-
Fr
0.00
N
}O.2
0.02
,2
--
Ol
I/
0.09
S0.00
.. 0.00
~~
N
0.01
0.19
. 0.30
$A
I ;
FRE0UENCY
F
-"
F,- 0.36
0,.
0.36
0rpodel.
1
2
FREQUENCY,
strip
tihe-ory
agreesi - v wilts botth the penis'it iseliid
and the
experiment
ol.i.
Ffs r ti eroll case thsre are some
inrerlved discrepasrciss tetween the present rielhod, strip
theory and exteriments.
. c orrerction
1 inally, it is shi.wn that tire snirtpl-sped
noriatilly used in strip theory does not adequately characlerize tIre actual speedeffects ote reed in tlie rseasuremenIs,
The sleed term in the free-surfacc coirlition has to IX.
included in order to obtain reasrable qantitative and
quilatitat ivc predicions. When the speed terms are included
in tie free-surface condition, it is shown thaI tlre present
resn Is agreequrite welt with exterimnsetal itata excepstirs
tire casesof the roll-damping and pitch-damping coefficients,
11seen hai the discrepancy in tihe predicted pitch-dampirg
coefficient of Series 60, ('e = 0. 7, hill form at a Froude
number of 0.20 may resulI front sone physical effects
which hi;,v- ii,been included in the present tnalheinatical
hi improve file predictions fir il pitch-damnping
siefiticienls consideration slould hi given it nrclusion if tile
c tI' rirdrl tcrisis twelwirli
rise tcillting potential and the
stcady ,t,urhrtin
tpotential which hirve lie negleclei, it
l~ile'
present callatioll,
0
0
mental data. It has also teen shown for this case that the
wave resistance curve predicted by the present method has
smaller "humps and "hollows" than those predicted by
thiss 'hip bco.
ALT
r
134
ACKNOWLEDGMENT
Discrepancies Between the Calculated and Measured Wavemaking of Hull Forns," Trans. North Fast Coast Inst. Eng
and Ship Builders. Vol. 67 1950.
16. Gerritsma, J. and W. Burkclman. "The Distribution
of .he llhdrodynamic Forces on a Heaving and Pitching Ship
Model in Still Water." Proc. of the 5th Symp. on Naval
Hydrodynamics, Bergen, Norway, 1964.
REFENIENCES
of Cy'linders in
I. Ursell, F., "On the Rolling Alotion
"ic Surface of a Fluid." Qtcrterly Journal of Mechanics and
Applied Mathematics, Vol. i. 1949.
2. iorvin-Kroukovsky, B.V., "hvestigation of Ship
Motkns in Regular Waves,"Transactions SNAME, Vol. 63,
1955,
3. Salvesen, N., E.O. Tuck and 0. Faltiosen, 'Ship
Motions and SeaLoads, "Transactions SNAME, Vol. 78, 1970.
4. Chtang, M-S. and P.'. Pien, "lit'drod.'natic lsrcc,
on a Body Moving Beneath a Free Sit/ace." Ist Int. ( lcronce on Computational Ship Hydrodynamics, 1975
5. Bai, K.J, and R.W. Yeung, "Numerical Solutions to
Free-Surface Flow Problems,'" Proc. of 10th Syrp. on Naval
Hydrodynamics, Cambridge, Mass., 1974.
6. Kim, W.D., "On the Hartsoic Oscillation of a
Rigid Body on a Free Surfice." Journal of Fluid Mechanics,
Vol. 21, 1965.
7. (hang, M-Sand P.C. Pien, ''Veocity Ftentialb of
Submerged Bodies Near a I'ee Surfac,-Application to WaveExcited Forces and Motions, " Proc. of I Ith Symp. on
Naval Hydrodynamics, London, England, 1976.
. Todd, F H., "Series 60, Methodical Experiments
with Models of Single-Screw Merchant Ships." David Taylor
Model Basin Report 1712, 1963.
9. Timman, R. and J.N. Newman, "The Coupled
Damping Coeffh'ients of a Symmetric Ship." Journal of
Ship Research, Vol, 5, March 1962.
10. Wehausen, J,V., and I.V. Iaitone, "Surface
Waves." Handbuch der Physik, Vol. 9, Springer Velag,
Berlin, 1460.
135
Abstract
infinity.
The potential function Y. can be
represented by a continuouE distribution
of single sourts ur the boundary surface S:
1
oj(a 1 , a2 , a).
'(x 1 , x2 , x3)=-.
x2 x3 ' a1,
a)dS
(j=1 ............. 7
(6)
where yj(x1, x2 , x3 , a1 , a 2, a3 ) =
= The Green's function of a source,
singular in a,, ag, a
a ,a
a
t
vctor, describi~g S2' 3=
.(a1, a,, a,= the complex
source streng~h
xj = je
= i...
,.6 (1)
is the aeplitude of motion
in which
in the jth mode and m the circular fre-
quency.
The motion variables x1 , x2 and x3
stand for the translations,
surge,sway
and heave, while x4 , x 5 and x6 denote
rotations around the OXI, OX2 and OX3
axis respectively.
The free surface at great distance
from the ship is defined by:
= r, eik(xI cos a ' x2 sin
Jo(R)d4
cosh C)xl + c)
+i 2 2(k2-v 2)
2 cosh k(a,+c)coshk (x c)
k d -v d +v
)-it
where:
(2)
in2_
T)2+ )x2-a 2 )
2
2
2
rl=/(xl-al) _(x 2 -a2 ) + (+
3 +2c+a 3 )
R=/,(x cai)2 + (x2_a2)
2
x2, x3
(7)
0 (kR)
t)=P(x,, x2 , x)e-itt
(3)
o
0
(9)
Y.(x
an j
,'1
,,a 2ao) dS
or
31
23 ....
for j = I.........6
for j = 7
n n1
= -
(16)
SO
where:
a kj= the added mass coefficient in the
bkj
(n, x)
n2 = cos (n, x2)
n3 = cos (n, x3 )
= cos
n5 = x3 n2 - xn 3
n6 = xln 2 - x2n
6
T
+6k)
(o)
rm
0
M
k
0
o
o0
m o
m
0
o
o
-0
0
where:
m
Ik
o
1
0
0
-I6 4 0
-146
o5
16
(18)
mode
Ik - product of inertia
k
Mean second order wave loads
In vector notation the mean second
order wave exciting forces and moments
about the fixed 0X1 X 2X 3 system of coordinate axes are as follows:
(12)
Subsequently, the first order wave exciting forces and moments can be found
T-2)#
from:
nk dS
= Xks
in which:
=wvexie
Xk
in thethk
force in
wave excited foc
=
mode
6
= phase angles
Cj, k
,o
2
{_2 (Mkjlakj) sin (wt+c.) +
b
W
t
+bki w cos (ut +1:)
+Ckj sin
t+r)4
k=1 . .
6
Je-lut
Xk -
(15)
n
n, through n6 are the generalized direc-
N4 = x 2 n 3 - x3n
WL
n dl __()Rdl
(13)
I2
P I
11 n dS
So
e-f
t
soi
nkk dS
-ff-((1) .VS(
1C
So
+
R(
(14)
138
) ndS
(19)
and
(2)=
n) dS
1 )
V(I).
-
+IR ('a(l)-g(x
) (x
5
n) dS
'
Model tests
The model tests were carried out in
the shallow water laboratory of the
Netherlands Ship Model Basin. This basin
has the following principal dimensions:
Length
210 m.
Breadth
15.75 m.
2
where the heavy bar indicates that the
time average has to be taken. The above
expressions are derived in the appendix.
In the above expression quantitiesm.
marked (1) are first order quantities
derived from the solution of the linear
problem described in the previous sec1()is composed of
tion. The potential
the first order potentials of the undisthe diffraction
wave,
incoming
turbed
potential and the potential due to body
(
Ir
motions. The relative wave height
is
at a point along the mean waterline
composed of contributions from the vertical motion of the point, and the potentials due to incoming waves, diffraction and body motions.
Water depth
moored between soft springs to determine the frequc -y response of the six
ship motions anu the mean longitudinal
and transverse wave drifting forces
and yawing moment. The test set-up is
shown in Fig 2.
- Tests with a captive model t- determine the frequency response of the
first
order
moments.
The wave
testexcited
set-up forces
is shownandin
The vessel
Calculations and model tests were
carried out for a lay barge type vessel
with the following main particulars.
150 m.
10 m3
Length
Draft
Displacement
KG
Roln
Pitch
Yaw
73750
10
20
39
39
1.0 m.
Fig 3.
Measurements. During the tests with
the free-floating model, the linear motions of the centre of gravity of the
vessel were measured by means of an optical tracking device following a point
m
m.
m
m.
m.
eral forces fore ani aft, and the horizontal rods containing the force transducers remain at a fixed distance fore
I aft of the centre of gra vity of the
vessel, the measured yawing moment applies to a vOLtLcal axis through G and
not to the vertical axis OX3 of the
fixed 0X1 X2 X 3 system of axes. In order
to make a proper comparison between caiculations and measurements, the calculations of the yawing moment are carried
out for the same axes. This means that
in eqn. (20) the last term is omitted.
During the tests with the captive model,
139
waves), the mean longitudinal and lateral forces ana yawing moment are shown in
Fig 29 through Fig 31. The calculated
mean longitudinal force predicts peak
values at roll and pitch resonance which
appear to be confirmed by the measurements. Some scattering 'f the measured
data occurs at higher frequencies.
The calculated mean lateral force
predicts a peak at roll resonance only.
Unfortunately, no measurements were
available at this frequency to confirm
this peak. Considering, however, the
overall agreement, the occurence of this
peak is felt to be realistic.
The experimental values of the mean
yawing moment in quartering waves (135c
are calculated from the difference between the lateral forces measured fore
and aft (see Fig 2). This yawing mcmeut
is small and consequently the accuracy
of the measurements is less for this
quantity.
In order to show the effect of the
four components in the mean wave drifting force given in eqn. (19), a breakdown of the mean longitudinal force in
head waves and the mean lateral force
in beam waves is given in Fig 33 and
Fig 34 respectively, The numerals I to
IV shown in these Figures refer to the
first, second, third and fourth terms
in equation (19). In both cases, it is
seen that the contribution due to the
relative wave height is dominant. The
contributions due to the product of motion and angular displacements have, as
may be expected, largest values when
there is a considerable amount of motion risponse. At higher frequencies
these contributions vanish and only the
relative wave height and the second order pressure due to the fluid velocity
remain. For frequencies tending to infinity the vessel acts as a vertical
wall. In this case the relative wave
height concribution is double the velocity contribution, the sign being opposite. This i!lconfirmed by the trend of
the calculations.
The calculated mean vertical drift
forces shown in Fig 35 and Fig 36 are
small and will not result in a signifi.cant change in draft of this vessel. In
the case of floating structures with
relatively small waterplane areas the
change of draft due to this force need
not Le insignificant as has bean found
from model tests.
Conclusions
The results of the investigation
have confirmed again that accurate predictiona can be made of the first order
motions in rejular waves of a floating
body by means of a three dimensiond)
singularity diztribution on the body
surface in its equilibrium position.
The method of direct integration
over the wettel part of the hull of all
contributions to the second order wave
exciting forces and moments leads to
140
11 Joseph, D.D.;
"Domain perturbations: the nigher
order theory of infinitesimal water
waves". Arch. Rational Mech. Anal.
Vol. 51 (295-303) 1973.
Newman, J.N.;
2 Dalzell, J.;
Application of the functional polynomial model to the added resistance
problem. Eleventh symposium on naval
hydrodynamics, London, 1976.
3 Boreel, L.J.;
"Wave action on large offshore struetures". Conference on offshore strutures, Inst. of Civil Engineers,
London, 1974.
4 Van Oortmerssen, G.;
"The motions of a ship in shallow
water". Ocean Engineering, Vol. 3,
pp 221-255, 1976.
5
John, F.;
"On the motion of floating bodies".
Comm. on pure and applied mathematics, Part I : 2, 1949, pp. 13-57;
Part II : 3, 1950, pp. 45-100.
10 Stoker, J.J.;
"Water waves".
Interscience publishers INC.,
New York, 1957.
References
1
Pinkster, J.A.;
"Low frequency second order wave
forces on vessels moored at sea".
Eleventh zymposium on naval hydrodyna;ic , London, 1976.
Salvesen, N.;
Second-order steady-state forces and
moments on surface ships in oblique
regular waves. Paper 22, International symposium on the dynamics of
marine vehicles and structures in
waves, London, 1974.
and
all
a12
a1 3
a21
a2 2
a2 3
a3 1
a3 2
a3 3
Co--ordinate systems
R = R
:12
~22
a1 a3 2
where
(2
x)
(5
-X
-X(
0
(26)
0
{
x()
x(1)
4
5
If the body is carrying out motions which
are a combination of first or~ttr motions
and small, low frequency motions induced
by the second order wave drifting forces,
the second order displacements follow from:
R(2)= R()
g
where
w
(2).
b
11
(2)
R
b2 1
b3
(27)
1
b
b2
22
23
b31
b32
in which:
b
b1 2
b
b13
- _ (x 51
2
2
_(x()+
x11)
x6
)
(2)+ x (x(
4
5
6
-
b3. -s
h
-3
( (1)2+
4
(22)
x 6
22
a(2)
&13
423
(2)x(x
5
4
6
-)+
b2 3
b3
(2)
(1
-- x 6 + x
b2 1
142
(24)
)+ R( ).
0
( )
x l
R (1)=
(21)
a3
]=
U 1:
a21
0)=1
L o
If the body is carrying out small amplitude motions the linearized (first
order) displacements follow from
= cos x5 c"':6
= sin x4 sinx 5 cosx 6 - CosX 4 sinx6
= cosx 4 sinx 5 cosx 6 + ainx 4 sinx 6
= cosx 5 sinx6
= sinx 4 sinx 5 sinx6 + cosx 4 coax 6
= cosx 4 sinx5 sinx 6 - sinx 4 coax 6
= - sinx 5
= sinx 4 coSx5
(23)
= coax 4 cosx 5
(1(2
5(29)
(28)
Fluid motions
The fluid domain is bounded by the free
surface, the surface of the body and the
sea floor. Assuming that the fluid is
inviscid, irrotational, homogeneous and
incompressible, the fluid motion may be
means of the velocity podescribed
tential by
rLl]:
(
(where:
=CO(ij+2 (2)+ ...
(30)
p ()=
The
areof defined
to
axes
fixed system
OX X X.relative
the potentials
The first order polential P"
consists of the sum of three potentials
associated with the undisturbed incoming
waves, the diffracted waves and waves due
to the first order body motion respectively:
) = 0
tt
3 -d)
(37)
g M-
1
_
(38)
4 (1)
p2= -gX
i
2((
())
_P,(2
t)-l
(39)
(31)
+()
0(1)
=()+
w()
w
d
bmine
Both the first and second order potential must satisfy the equation of continulty within the fluid domain and to
tirst and second order respectively the
boundary condition on the moving surface of the body and the fixed horizontal surface of the sea-floor.
The boundary condition at the mean
free surface becomes:
(x
3
-pg(X
p(1) =
(32)
gO(2) (2)
+
= _24(1).5(I)
2(
t
tt
x3
+%0 ( (g) ) + !I.(
g t x3x3 g ttx 3
F = -ff p N dS
(40)
S
where S is the instantaneous wetted surface and N is the instantaneous normal
vector of the surface element iS relative to the fixed system of axes and p
is the pressure given in eqn. 36.
Since the budy is moving in all six
degrees of freedom, N is also an oscillating quantity of the following form:
-(o)
-(1)
2-)2)
(41)
N=N
+rN
+
= -f! p N dS - if p N dS
(44)
S
s
0
Substitution of the pressure p a2 given
in eqn 36 and the normal vector N given
in eqn. 41 gives:
11)
where.
- g(X -d)n
3O
-tf-n((1)
I'))dS+M.R(1).R
SS
(53)
where the heavy bar indicates the time
average of )he quantity under the bar.
(47)
so
2 d
o(-
S
0-
(48)
2d
S
0
t 1
)ds_;_m6(2)cdS
t~
t
t
O
WL
_ifr~(().v6(i
3 ndS
in which:
21
dS
(49)
R(2)
xx(2
x
xI
g -
( )1
0
(xx )ds-!..'-_p~
(R 1l: 1)
X5o
II
4
WL
.R(1)
.
( ri)dS
()- X(1)
q
(1)
(54)
where:
1
Ieo.
4
4
0
15
-1
o
64
1 46
46
U
1
(55)
." l
tarmtinvner ng Ilar
athe eerto
ac,
(|tf
is according to eqn. 26
rectoYTnd R(1
and F
according to eqn. 51.
Nomenclature
()
where:
M.
_ Spg (
frequency in rad/sec.
[0i0m 0
,,
(5)ols
oo
and by
nare
the !irst ordernotions of
the bogy under the Influence of the first
i(t)
a
144
2(<)
g
x
R
(1)
k,
R
R
R(o)
X(k)
2
(
(2k
m
14'15,16
146'164
(1)
and OX
axes respectivel
Ot
the fied system of axes
mean second order wave exci-
to
k
V
L
B
d
145
x3
x3
GI
Xl"X
5 METRES-'
5 METRES
150
LENGTH
METRES
50 METRES
10 METRES
BREADTH
DRAFT
x2
. . ..
..
..
. ..
X2, 12
180*
G
BALL-JOINT-
-FORCE
TRANSDUCER
135
-SPRING
1go.
FIG.
2:
140
\BARGE
\ RIGID 6-COMPONENT
FORCE TRANSDUCER
~+X2
FIG,
3:
|'FORCE TRANSDUCER
~BARGE
147
9
COMPUTED
MSMEASUREDURED
---
--
1.2
MMU
0,6 ..
070
1.5
4,5
3
. 30
4.5
3.0
VERTICAL
FIG. 5: AMPLITUDE OF
0
DIRECTION
WAVE
FORCE
WAVE
AMPLITUDE OF TRANSVERSE
FORE
90A
WAVE DIRECTION
FIG.
1,5
0--
,-
0D---
WAVE
COMPUTED
018
MEASURED
M
COMPUTED
MEASURED
0.6
0.12
--
0
-~
0 -
15 10
3.0
1.5
3,0
MOMENT
FIG, 6: AMPLITUDE OF ROLL
WAVE DIRECTION 90*
148
4.5
WAVE
18
COMPUTED
MEASU ED
1.
1,2
01
COMPUTED
MEASURED
06-3.
9:
FIG.
1.8
0.18
-___
* MEASURED
0.2---
COMPUTED
COMP'UTED
MEASURED
1.2
Jo-Jo
--
0 0---
a.
FIG.
10:
4.5-~-
ci0
A0
115
4.5
FIG.
149
11:
3.0
0.18
0,9
//
/
0.12 -
------ COMPUTED
.
MEASURED
COMPUTED
MEASURED
0.6
)x
15wVT7 3.0
o
12:
FIG,
5 wV1 1 3.0
4.0
FIG,
135*
13:
4.5
180'
1.8
COMPUTED
MEASURED
COMPUTED
MEASURED
1.2
__-
0*
>
oh
LA
x
00
ol
FIG,
1.W,5
14:
3.0
4.5
1.5
WIJ-Ug
3.0
150
4.5
181,8
COMPUTED
COMPUTED
MEASURED
MEASURED
1,
L2
1.2
L,2.
:\
" 0,6
. oS
1.5
0.6
4.5
3.0
FIG.
17:
1,8
COMPUTED
*
MEASURED
4.5
WAVE DIRECTION 90
COMPUTED
3.07~
1.5
MEASURED
0
3
--
I
0
1.5 w\/i'-
--
-0.6-----
----
-----
.
"
3.0
4,5
1,5
3.0
151
135"
4.5
1.8
1.8COMPUTED
COMPUTED
MEASURED
M
MEASURED
1.2 -
1.2
0
xm
01
1.5
WI,-
1.5
4.5
3.0
21:
FIG,
W\F-g
3.0
1.8-
COMPUTED
MEASURED
COMPUTED
MEASURED
1.2
Ci
15
FIG.
22:
30
1.5
4.5
FIG,
152
23:
3,0
4.5
1OMU8E
0.9
S
COMPUTED
*
MEASURED
MEASURED
1.2
0.6
00
~-
01
1.5 WVLg3.0
FIG.
0.3
24:
FIG.
1.8
53,
4.5
25:
CMUE
COPTD1.8
________COMPTEDCOMPUTED
MEASURED
CoMEASRED
MESUE
1.2
1.2
to
0.L6-
o
FIG.
26:
0
3 .~
0.6-
FIs. 27:
1.5
3.0
153
45
18PUE
-3.6
TED
--COMP
*
MEASURED
12
CMUE
COMPUTED
-2.4
m-j
Pm
a.
0.
3,0L-/ 4.5
FIG.
28:
0
FIG.
18
1.5 wpV-
29:
3.0
2A4
MPUTEDD
* MASURED
--
COMPUTE
MEASURED
121.
K'O
0
FIG.
1.5
30:
FIG,
154
31:
4.5
-3.6
S
-2'4
COMPUTED
MEASURED
--
0.
lb
4.5
32:
FIG.
60
30
66
-6-1
--
-I- lI
0-
.......
IV
IVL
-30
FIG.
33:
a1.5 wV-i7
_
3.0
90"
4.5
-1?0
FIG.
,.
34:
3.VG
0
180'
45s
3COMPUTED
60COMPUTED
30
:N
-3
""PUT
FIG.
36:
X3 X
X3
PLANE
VERTICAL PLANE
-"I
FIG. 37:
AND G, x1
X2, X;
X3; G,X1,
X2 , x3
DISCUSSIONS
of three paper'
Invited Discussion
F. Ursell
University of Manchester
Discussion
E'y
Rnan
of paper by M.S. Chang
lne results shown in this paper are very
impressive, but not too surprising. It is well
known that the strip theory gives good predictions of ship motion characteristics only for
high frequencies, and for zero speed the comparisons shown here with exact three-dimensional
calculations are qualitatively similar to various
previous studies. The use of the strip-theory
forward-speed corrections in conjunction with
the three-dimensional zero-speed damping and
added-ma s coefficients is inconsistent, and the
resulting poor comparison with experiments is
to be xnected. But, Dr. Chang is to be congratulated for going on to solve the forwardspeed problem with the correct free-surface
condition and source potential. The resulting
agreement with exper~ments is generally satis-
IS
Discussion
by H. Maruo
of paper by M.S. Chang
Discussion
by V.Kayo
of paper by P. Guevel, G. Delhomeau ano
J.P. Cordonnier
*=
f32
a20
X2
+i4'+Kz)Ja,(x')N(Klx-x'l)sgn(x
J
-x ')d
x,
where
0
N(U) =-y - ',n2u
+ 1 H,(u')du'
2u +
X2
on z = 0 inside the body.
u
vi J(u')du'.
fo
,(2D)is the two-dimensional solution for an
oscillating cylinder and ao(x) is the source term
in the expansion of the two-dimensional potential. The boundary-value problem on the hull
surface is formulated in the form
i(2D) "U()
- W(.)jn,
Therefore thewave-
+jJo(u')d
'n
where U(x) is the relative vertical velocity of
a section anJ n. the z-component of the direction cosines of the normal. It can be easily
understood that the three-dimensional effect
appears in the function W(x) which is similar to
the induced velocity of wing theory. I calculatedmodel.
a numerical
example
aadded
Serissmss
60, C
0.7
The result
forfor
the
e
H(o,K
sec
a)
f Fa izso
2
ote-a I s sa
the
dobaprbimy
potential
us to be
asbed
the zero-order
approxmeewhich isis
read
at the joint
eeting
of the three Japanese Societies of Naval Architects in November 1977.
Discussion
byYKusaka
of paper by P. Guevel, G. Delhommeau and
J.P. Cordonnier
The discusser wishes to express his respects
to the authors for their efforts to obtain a
numerical solution of the Neumann-Kelvin problem.
It took more than 10 hours of computer time for
thediscusser to obtain only one solution of
this problem using the very large and high-speed
computer HITAC 8800/8700. As the authors also
point out in their paper, it's necessary to find
out the new formulation of the influence coefficients. However, from the discusser's experience, the interpolation method seems very
dangerous because of the oscillatory characteristic of that coefficient.
fS ,+ SF D(Q)K(P.Q)dS(Q);P ,csB+S Fi
D(Q)K(PQ)dS(Q); PQcS8
(2)
Is..
ABSTRACT
This p3per presents a novel integralequation technique for solving the steadystate wava-resistance problem. The freesurface condition in linearized, but the body
condition is satisfied exactly. An integral
relation describing the flow inside an
arbitrarily truncated internal region is first
obtained by applying Green's Theorem, using
only thz simple source function for an
infinite fluid. The internal flow is next
matched with eigen expansions in the npstream
and do wtream outer regions. The radiation
and
ouer
owntrea
egios.
he
rdiaion
importance of nonlinearities
onition hen
fra-curf
Bointed the
with the
free-surface
vhen
the conventional
Froude
number iscondition
large. But,
(add,
I. INTRODUCTION
The practical importance of the problem
of predicting the lift and drag of a body
moving in or near a free surface in well
known. Much attention has been devoted
to the subject matter in recent literature,
This paper describes a novel integral-equnttion
formulation for tackling the steady-atate
problem with a linearized free-serf ive but
exact body boundary condition. Emphasis iS
placed on validating the numerical formulation
by applying it to obtalttflow solutions for
various two-dimensional bodies. The formulation permits the body to have cirt'nlatior,if
present. The extension of the pre,-snt method
to three-dimensional problems w!illAlo be
discussed.
A brief review of the literature
pertinent to techniques used in solving such
steady-flow problems i% included here. By
expanding the solution in an infinite series
of wave singularities which satlify the treesurface and r4diatlov condittons, iavelock
(1936) obtained the wave resistance and lift
force on a circular cylinder in a uniform
streas. His formulation permits the body
boundary condition to he atisttfed exactly.
Tile problem of the fle. about an arbitrary
150
motion aO
distvrbed
tiLeuistu'
the motlov
-in2 LU:1Z--t.,
sutzee. Lct
.. P-,
t
ry.
-
(x,yt) be
I t due
t' - C
Th,
,at
.i)
inematic
ed on the body
(2.7)
-.-
whe'
U
unit -xterl
oedition I
--ed
the body and n a
to
e fluid. The bottom
a, nfic. is
0
f(
(2.3)
kn this m
iced ft-
fOrm
4
U4.+X'(x'n~t)
_
+ gyff
(2.4)
Ir
YY--h
Figure 1.
Coordioste System
and tttatooe
Y--h
SFi
Kcx (X,o) +
U2
0,
(2.5)
isoviously'unneceasary.
The physical quantities that are of interest In the steady-state problem are the
ree-surfacc elevation n(x) . wave-resiotane
1and lift force L . By the use of
equation, these can be written
In terms of the potential as
S
[c0(1~
C- + 0 1)
a
5553ernoulli's
+
asx-+
1C+~y+. W
(.)
(2.10)
nx - U 0(X.o)
where
C
-P.
4~+-(.1
p(2U.11)EF
2
+-1
I'
_'K
3.
(2.8)
K
~(bs+#)
TE.C
d
d
(2.12)
THE HYBRIDINTEGRAL-EQUATION
TECHNIqUE,
I (Ux+*),jppor, -
~i,!I~'
Lx2
where 5
is the dynamic pressure coefficient.
Note tile
L as defited Ine not incluo,~ tile
hydrostatic force.
is
-(2.7)
Kt
g fF1dy
0
ts +.;
K
(2.wer.
#oei
L -
(29
162
!)~'d--i
(0.1'
2wo(P) -
nlog r d.
n log r do
log(rr') +
-d(P) +
SF
I+
log(rr') +
1
nlog(rr') ds
(3.2)
+(,.nlog(rr')
4+
r fnlog(rr')d
S+
K
so
where
n-)
r + Kioxlog "I
~~ On~lo ~ido ~
~ ~.lgr-
E+uE-
foL
log r di +
SB J
(3.4)
1n
o
P E S
S
0
with ((,n)
being the variables of integration along S . Now, by substituting
the conditions (2.2), (2.3), (2.5), (2.7),
and (2.8) into (3.2),we obtain
2ro(P)-j
-Uj n x log(rr')ds ,
L- log r ds
+a(x,y)
(y+h )
0 a
i
logr+U
f0 nxlogrdo
on
(A comx+B sninmx)cSh
cosh moh
(3.3)
*+
+ i Ck o'
k-i
t
for x < x
log
-E)
-'
+-4
T
7K
p,
t on
(3.5)
- tanh oh
y-h .
reby, if
(3.1 reduc.. to
rcos mk(y+h)
where
-
represents either
ma
(3.6)
or
imk
with
f
1. The values of the coeffilaents A,B,
Ad C, , k-O,l,2.
.'. . are unknown and have
to bt deterMined from the inner region via
S ,
le3
lo~r')
c
I~-lo~rr)
-m~P)+
~'
ds
f-log(rr')
Rdegrees
Clg~rr)ds
log(rr') da
E_
" C a
logc')d
-~
xa
si1o
" B
Lu in
+
kit
ocsiiX
1
no
k(-k+ k
kk)
(F+C k
kb
-U n
f~
j n
0J
o( r)d.pvs
d*
oWc)
nApedxA
Diac etization of Integral Equation
17n3.2
qc
so
Cjlo
ie
log(rr') do
da + r
ftn
(,I
))*xo-
So
Fu,, G 0
Fk
(3.10)
N
where
are
F.
-l(
p)
di .o
i
ao log, rr
+~af .Y)
epc
0.(
(3 .y) sk the,,
oftthe
procedure
for thn
problem.
prm
s.
n
eqacte
ae
*V
fcus
eotsdifretatoni
roato
of) thean
(3.r9)a
164
fuolution
nnsn
ror
)'0
QjXjX6.Y)00
_In.)p
"'1J-1
(3.15)
q,(x
fo'
here
IM
(1.YE0,0c
dx+
(3.16)
.rc.
h"
I -
ko,
j-1, 2,
.~ no)
poInt! defir.inj S
of,
denotes
With the
,f (3.13) and (3.15), ( 3.7) can nov be
;ed in terms of a finite number of unas follows:
syet
F!
S Fe'.-r-
( O1 Aj)
di
de
r)Llogfrr'
""
+
.n+i
(i
J1
FF
F ,v'+
F,
r-h
oa +3s~~
tqio)~~~~~~
:11
Inlgrr
r]
+ CQ~xyc~yx
Q)~
(3.14)
+
-'
Ccos mox)
, Gik+
(n ) F
I0
+Gk
for any P C So
I-i
On the btxindary S
, which will he
rapr seated b'. stralght-llne segments a, in
Yeang (1975). a scheme Involving aid-point
t imued. 'This enable.
derotisation of
as to write tit third integral of (3.7) and
the right-hand side (3.7) respectively as
(0.17)
)(ore, the iiumlirr 0l terms tted in the etj %l
expansions aire derosted by N- and #1+ , for
outv-rregions
the upstream and doowostream
respect ively. The, integral 0) Q.bkyeLa that
defined by 0.16).
108
At this point, we inspect how the unknowns are distributed along the boundary.
They are listed as follows:
Number of Unknowns
on
SF:
N+2
S:
No
N++3
S;
grid point
w-field point
(-(0
doxxn
')
for n- ,1,
(3.18)
-R7-
1- ;Tnh 61oh
(3.19)
where
Ct C
I~
- 0
L
1h 1
r~ I EhJh(I-F')
+ r(Y
(,x+)nxdo +
"( h
SoUS
(1.20)
We note in pasing that one of the aquations obtained by applying (3.17) in the
manner described above will be redundant. This
does not seem so surprising if we recall that
the boundary-value problem stated in 5 2 can
only be determined up to an arbitrary constant.
Interestingly enough, the redundancy is of an
implicit type for no one parti.:ularfield
point is more preferable then the others,
However, this redundancy can he easily removed
by assigning an arbitrary non-sero value to
Co- (or C0+) during the reduction stage of the
ISO
r),
(x,y,t).,r
(s)
x.y)eit
(r), AYen
mX --.vh o
or)..A
r
h
c(8)+., -ei xh
time now
and correspond
the spatialto functions
ar)
nd e*(s)
fictitiousI
rsdiation and scattering potentials respective-C(s)
ly. Each of them satisfies (2.1). (2.3) and
(2.5), but
(
d/a-2
Fh-0.8
981
0.5
0.5
0.0
-0.041175
-S.35281
X .1
01
58
0.00778
0..002
0.31290
0.40805
-.
8.*1
-,.
1 +SO
-1.45080
-h.55908
0+.51181
-.
-0.ins
.or
Ot?
OS
.i*i
-:
0.08089
-0.2700
-3.30831
-0.2173
-1.55881
-I.0180
-0.0041
-8.70808
-0.O009
-8.7285)
00
04
-0,1m1OS)
-0,88118
-I.
?1750
-8.02018
-1. 28053
-0.508819
-1. 88010
-7,25078
95
-0.0118
-2.07101
-1.30118
00
-.
-.
10
101"
-0.00800
0.0212
0.072S
0.0 12
0.28520
1
10
73
0.4002
1.
0.08110
ILOC8I01C0078T
.074
-0.
00080
-0.38920
l|
a0.0010
-0,01928
0.0328
4
0
8.802
-o.ssOl
4.88)88
-0.80082
-1.08033
-0.09147
-.1.02O
-..2....
-8.88808
.8.2,75
2 o
-3.70399
-.. 81858
..
,.......
0..
......
.
0
.....
.........
o...
....
0
0.3050
-.
.0.+0l,1
-08
o.
0
$
s,.
-. 0)825
-O0l|20,1m~
0.00117
-osaoo8
-,0I92?8
i
I8
cu
-
-.. 233
-0.92110
-5.,200]-8.54.02
-. 099
-1.
IF018
-0.91802
-2.00000
-0772
- CO#
V . 1.
.....
o.T.i
..
8
0020
-0. 80000
-0.107)1
-4,0.
80
-. 1106
-1,8184
-2.50 A4
-0.20020
-a.
30820
88
00
6.01852
0.0089
0.24059
-0.48500
-1.smsn6
-. 5081
-3.35885
3,120
0.08113
-0.14005
0l
os
58
T
*0.!&flop
-5.0088
-S.2410
~
X'
0'T
$$
CM
0.08803
0.0852
0.0soon
-1.441
SLO
OIr
808v840nW
-~.2..69
-S.30 '
-5.IsOo
5.0
0.10000
asx-
h/a-5
08 FREE Sw290ct
I - 5.80000
"
-. om
S
-0.
200
0 haX
a4,)
Is x cosb a,(+h)
os)m
x csh may8
0oeh
cosh m h
whereas
-un,
(
+ Te
as N + wo
8o
50
s,0$01
-.058a
-0.05209
0.98881
-0.93859
8
5
-0.O01s1
0.023
.50
l
*I11
Table I
Coparison cf Solutions Based on seal Formuleti-n
and Complex (-ctms) Pormulacion
187
..............
+ .o
0.3009
0,Is0" q
0.0)
.802
-7.088150
0.984
V1 ''.
*2..l.0
5/4.3.
520)04
ios-svro. ,"'t'
./ ,u--1.4 .0133
-1,0 .. 1.,
-O.S .37ms
.33
5.
the relation
o7
.2416
0.5
1.0 -.
(r)
.d -d4
2d-
(s)
.0112
.90
.1218
.3511
W13
.. s07
is
-.
-6.207
.
.241l
.711.
-.-
.o -.
."33
.1343
.1244
.3912
-.
..V3.9.
6419
002-.
Y .00.7 -0.)437
7.4603
-0.415
0.3100
.3134
4.311
0
-.
.7773
-.0.5
.06
-. M011
-.00
.W071
.7100
.0316
-. 5
-,0005
.3246
-..40
.,0010
t.,.,a
-. n
-. 0,6t0
3,-11,2
-.043
c-.-'
4
a-3.d
(,4
.075
c'
'.-'
0077
.326
,,00,4
.oon
-.0750
2
.oo'
Table 2.
Solution of Flow shout Circular4 " Cylinder
for Different Values of y and F-
Next, a uniqueness check on the coefficients of the eigen expansion was conducted
in the following fashion. The problem of the
flow about a circular cylinder is solved thrice
with three different locations of the radiation
boundaries, E + and E - . A correct formulation should yield the same solution regardless
of the location of these boundaries. Table 2
Is a collection of the numerical cesults
obtained by using the firmulation in 3 for
xf /a =. .2, - 4, t8, where a l the radius
of the cylinder. Note the excellent agreement
of the predicted free-surface elevation
n(x)/d among all 3 cases. The coefficients of
the eigen expansion are also quite consistent,
although the s'curacy of the higher-order coefficlents in the expansion tend to detertoiati,
as x+ or x- becomes large. In view of
the fact that the solution in the outer region
is always dominated by the first few terms,
clueto the exponential decaying factor In
front of CU In (3,5), such inaccuracies
hsve little overall etrect .2 the nouIutil'.
le
Li"V-772
dM- Rk -ASS)dla-Z
Wd-14
LI
-DW0.7
1.4 Wd-
1.204
.2
(h
0.4
0.5
0.4
-0.5
0.0
L.
3a.
(IPYUh
Ce C;0.3
3b. -Supercritical
-Subcritical
Figure 3.
0.5
da
21
0.4
S0.2
---
0.2
03
C14
MS
0.6
Fh
0.7
125
O.B
0.9
1.0
0.5
-----
d/a - 2
h/d - 2
0.0-
so
1.,5
0.2
0.4
0.6
0.6
1.0
1.2
.4
1.6
169
1.B
number. When nondimensionalized in the conventional manner, both lift and drag coefficients are observed to have turning
points in the subcritical range. For most of
the range of the Froude number, the lift
coefficient Is considerably lover than the
inflnite-fluid value of 1.08
S. CONCLUSION
In this paper, a novel Integral-equation
method of solving the steady-state shim-wave
problem with the linearized free-surface and
exact-body condition is presented. The method
is tested for lifting- and non-lifting flows
about a number of two dimensional bodies.
Numerical results obtained using our method
agree very well with existing calculations.
The formulation incorporates a rational, yet
remarkably simple, treatment of the radiation
condition. The current investigation shall
provide a sound mathematical basis for
tackling the more practical three-dimensional
problem. Such an extension is conceptually
straightforward. The source function will
be the simple three-dimensional Rankine source,
1/11. The contour integrals will now be
replaced by surface integrals. Extensions
using similar techniques have already been
carried otitsuicessfully for the time-barmonic problems (Yeung, 1973). The present
mathematical formulation should provide
additional insight into the understanding of
the Neumann-Kelvin ship-wave problem. Since
our approach does not utilize the traditional
Havelock sourc, .unction,it will allow us
to bypass the controversial issue of what
the rational treatment of the "line Integral"
(see grard, 1972) around the ship hull is.
For a body with circulation, our forzulation was tested by calculating the flow
about a NACA 4412 hydrofoil submerged at one
chord length from the undisturbed free
surface. Figure 5 shows the pressure coefficient on both surfaces of the airfoil,
Only 34 segments were used to represent the
foil geometry. The gutta condition (2.9)
was handled by a three-poit finite-difference
scheme and was evaluated at the field points
adjacent to the Lcciling edge. The results
are t.,good agreement with those of Glesing
and Smith (1967), particularly in view of the
sensitivity of the solution to a precise treatment cf the Kutta condition (see Hess, 1975).
Figure 6 shown the wave profiles generated by
such a hydrofoil translating in water of depth
equal to four times the chord length. It is
interesting to note that the initial
depression of the free surface moves further
upstream as the speed decreases in subcritical
flow. In the supercritical case, the disturhance generated by the bitdvis felt at a
much larger distance upstream. Figure 7 show;
the hydrodynamic lilt and drag coefficient of
the same hydrofoil versus the depth Froade
t.s
10ww suace
Cp
0.0
-0.2
Os
0.44
0.
rWMt
Ging
G4 Smth (1967)
,- F;41
d/1 - 10
CL.756
0.9
.-.
-1.0
-0.9
. 8 -0.?
-0.6
0.5 -0.4
06s
-0n -0,1.0.1
0.0
170
-. 5
f-
d1 - I
1.2
07~
Q~r
'.rure
~03
We ou
t&AkDT
dC,c.
lke Not .J.lka
aMRC.nmi
Prfts..r
,f MI
of a
1 Hydrofoil at
ofAbramouttz.
f~r
rng a. Na'oe
'9,btelu.,
l.A. NMbok
ofhwia
Standord.,.
Na9b of
a
AT.
fSadrs
94
94l. K. J1. A l
F~llo
initr-F.rmet Method
for Steed %~-lomwsomal
py@#Surface PTm Probia., lot
.Cy'
Ii
or
..
'7.1570
Sa.
I1
. I
A 'oa
Fl.. j st,-le.sr
hetlod
frFr~ a-lt
bstlDis~o,.
&I Hv,rofoll*a R'aos.tpr for .1.ship
Veung, R. W.
Chan, R. K.; Stuhmiller, J.h. Nume rcal Solutnon of Unsteady Ship-Wave Problems.
11th S
Navp.N,,al Hydrodun., London,
1976, 303-313.
1975, 581-608.
Me.A.,
Vol.
'R (1969),
A Hybrid Integral-Equation
for
Ii1-
432.
Tuck, E. 0. The Effect ot Non-Linearity at the
Free Surface on Flow paNt a Submerged
Cyl nder. -% Fluid Mech., Vol. 22
(1965).401-414
Iehausen, .J. V.; Laitone, E.V. Surface Waves.
"Kandhuch der Physik," Vol. IX, pp.
446-178.
Springer-Verlag, Berlin, 1460.
172
APPENDIX A
L-
Jm
7(x-1,y)
defined
and
by
i,
Fk and
The Integrals
log r
w(z)
+ Jnsinh mr+ih---)
i {sinh(inh)iog
--
dt.}
(A)7
sin(mh) log z + i
log r
do
m(z+ih)
-h
-m z
a -
y.
ly)
- MZ
(z+fh)
G(
(A.4)
(A.4) in the u-
J M(
u)
p + i ,
u-- n
-M(z+ih)
z =
-m(z+ih)
d , cosh m( +th) m
=CCt
t;(z)
log(&i}
Fig. A.2
tA.)
W(t)
7H'.
ro
is a
;)mown
-0 as
FI,)
0-
5)
n-a+lv
(A.1l)
dt
, jarg zt<v
I-E,(af) + E,(m(a+ih))
ez
.-m(rtilh)
-I
[d.
--
(A.5)
sin(Mh)log I
Fig. A.1
-I(.YS,
1(z)
-lb
El(z)
ponential integral
.d(AA)
- 2r)
I-r(-z) + Ej(-m(+ih))]
Whence,
Fe
1
2
z t )
em
z - a - t(2h+y)
[ l (z
tI(mz) -E,(m(z+ih))]
f
-e m~z
[)[E,(-mz)
-E(-m(z+ih))]
-which
(A.7)
APPENDIX B
Relation Between Blockage Constant
and Body Potential
(A.8)
+ +
.(csy)
(A.13)
e(
h)(z+ih)
E(m(+ih)) -E (mz) ]
+-m
(-m(h)[
i)
1
[ 1 (-m(z+Ih))
-E (-mx)]
(A.9)
(x 4,y)-C
Note that as G-0
+ ON(xl)sin (mx + 6)
the exponential-integral
m(yh)
cosh Moh
0
0.1)
El(z*) - [EI(z)]*
(A.1O)
f(tlt
cos m(y+h) ,
(A.11)
i
sF
x[4x(x
x+
(B.2)
Ur
2.*Id-UhC-)(7-)
(.3
ul"
Qh1oglyl
Next. we note that because of the body canditlon (2.2).
-coo
*An)d
0(y+h) [si(a(y+h))
If '
SiC-say)j
(A.12)
-#,;')da
SOUSI
(3.0
where (V.+Ve)
is the 8a8 of the aubrgmd
area of the Ldy and the net protruded area
174
d
-urJnda
ur(h+YT.)
5)
(B.
SoRS
(i.6)
Abstract
IX
1. IntroducLion
In earlier numerical work on the circle
we expressed the potential as the sum of a
wave source and multiroles, which are simple
but specially appropriate only for the circle.
For more general shapes the method of integral
equations may be use-. In our work, Creen's
theorem is applied to the potential and to a
fundamental solution (wave source). If the
fundamental solution satisfies the boundary
conditions at the free surface, at the bottom
and at infinity, but not necessarily on the
body, then the resulting integral equation has
a complicated kernel but involves only the
values of the potontial on the body. This
2 2.)
(2.1)
*(x,y,tj
jy
t(xy,t) - U
-xp(-iot
In
on the
46e
(2.3)
(2.4)
KO =
Uocexp (iut)
.X
D2
K(P+
X' X2
y0
sD
24,
isa
)24,
ry
4x
= 0
=0~
y=h
Fiqure 1.
i(xThe
0)i(x,y.)
+
-
0 as x
(2.5)
",
coshk(h-y)cosh kK
kb-k sib kh
+
+
-kh
e
sioh ky sinh
k
nsk- --)
k
J rush
log
iY"
2
x-0), + (Y+v)
(s!y;v), say.
(2.8)
y-,)2
lyvr)2
2 +
Jf (my)
j log
G(x,y;,n)
(2.6)
tanh kh.
Cirr,.E
der r - a, 1i
SL
I
xy;
j
eda) +
dO
-(0
-al
co
on 0
-"
(7.0)
177
Ot
log) +~
>
-0
%#(a
-Ut
-K) avelocity
(Kh)
s~l
(Kh)
2s
7r~con2s9*
)~!
I(cf.
~2-1
WY,
.
k___
k
_____h-
--------
where
C2~
(Kb)
2@
WV.,;
e~remainder
e
(L.12)
(ho,2
(K)
TkF
"d
i
(-1.13)
~mm n
Kh) G o. (fh)-C;
(Kh),
(,)-G(P,Q) - A f;(P,0)G(Q,t0)
2sal
2
4i*) - co( s.l)1
Iare
Ah
(2sn)!
e..*~
(Kb)
o
B-00d
26f
1)Cs o
28+ (Kh)
~~,
-c
-
'o,
jc
~an
-(-Kr*)
4Khi
ilsh 0
( ts)
sin .0*
~*
da
Co~0
(2.14)
G2
(.0
logKr*
1G*(~y;Cn)
()
(3.1)
~.
ii:-.~uslti
'IM
of the kernel of
4(P)
f(q)G(P,q)ds
DDI
TP(p) +
f
a(pq)dsP
5 (q)-L
an
q
3.3)
If we write
(q)G(q,O)ds
(3.4)
((x,y)
51)
--
A 2C (k0 h )
2i1-11Q[v1
0,
( 4. 1
(3 .5)
dD
(4.1)
2
0--,
I (Vx)} dx
by definition,
3D 2
2
4. cosh k h
2kTh + sinh 2kh
Put A
Then if
(3 .6)
niA
where A and A 2 are real.
and i2 do not fie on the circle
A,
S
A2
2 - 6I
~C
QV)
)
(4.2)
C3.7)
(j)
5. Additional modifications
Foforcom.ntof cmery
v(o)
.d
(-
-- atlcoso
Cl
1
* U
rooh (tO.a)vG,(-.i)
4
119
aqt(on
(5.1)
a dO.
and
I
PM
6K(,0)(0)dB
= cos 1
(5.6)
J
-IV
The numerical solution of (5.6), together with
the evaluation of (5.4). took approximately
one-third of the computational time required
for the solution of (5.5).
(5.2)
d
(virtual mass
ddKaro
as a
u
{-ioU0exp(-iot).(,a v)
(5.3)
Re
dO,
S- JeF(6hyO)ds,
O,
(6.1)
2J ipotRe<1)exp(-iot)cosO a dO
-
a In
itency
f
2J].ioo{Re<>)exp(-ict)cos0 a dO, whence
virtual mas
1
L46
d
""
(rl
vr
4
l
lh h b
4.. a,
r,,uetl
(5.4)
where
KhI -h,
*(s)
Ca)
J07.
(O,o)$(0)d5 - F(u)
(5.5)
Numerical ne ode and results
Two ntwurrical aathods ware employed to
tegral equation
solve the
10
Then
K(O,a)*(e)de
0(0)0
A!
A*O - n;
(7.1)
=F(ai
i,j-l,2 .... , n.
Also write
BO .
(7.9)
-B..
i,j-l,2,.n.
Mi
2
CO--C;
C.
*-
O~d-F()
() +is (O~a
L~l -
(.2)
~ap0. ~ 5 ()
n~
n(o) - F(s) + a
i*0,l,2,..,n(7.lO)
such hat
LYj
0[n
+ B..1 - -C.,
(a(73
(7.4)
0.
(ii)
gives
LiY
du
aloihm
to-
K(6a~.(4d9~equation
(7.1)to a system of linear equ~ations
5f~.()
J
by employing an elementary trapezium rule to
evaluate the integral term:
b
IaKen40d
-
ehd
.. the integral
This second method rred,
,b
n
olyn'
qua~dratr
(73O(aih)
~'(avih.
a~jh)4(sih)
-,,
wherei
(7.6)
j-O
(7.12)
whencej0
aand
-c.,
(lA*.+B.
13
3 i3
i-),1,2._n,
(7.7)
1j
A t
where
~,Fand
* J . Ibwhere
- f:i65)~~dsrespectively.
O~~j.lement
-JK().a)4
jjb
(7.13)
E + t(J)
A is the msatrix
are AKmob). M(ah)and F. .
lIK**
(7.8)
a.-IhK(auh,a)
h~~
(4b
l-hk(a~h.ashl )
.- JhKt(a~n,h)
iaA
For the circle, a - -J, and 6
Js; ch,)ooe
(.4
-igha
4 WiChah
-1coos2..
1ei
8. Conclusions
A (
h E(
)
(7.15)
E
h E
Sh
-the
(;) +
(7.16)
(Cm)
oinstabilities
The i
element E.' ' of the colun vector
-(j)
is given by Gregory's finite-difference
formula (cf. Jeffreys and Jeffreys, 9.083)
th
Ei0) -+
I- 1
I_
Ca)
(a+ih,a)00C)
K..)
(and
(7.17)
Method
Tim,
We should point out that our values of the
virtual mss of the circle fox low wavenumbers
and shallow water (a/h , 0.4) exceed the upper
bound given by gal (1977) for sero frequency.
__-
bcr
2OT
)Toccur.
6T
Table 1.
Circle,
a/h - 0.5
VIRTUAL MASS
Ka
0.00001
0.00005
0.0001.
0.0005
0.001
0.005
0.01
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.822
2.5
3.289
4.891
5.0
10.0
Multipole
Metho
Method
Modified
Integral Equation, A-I:
W Galerkin Method
0.49843
0.49845
0.49848
0.49851
0.49855
0.49880
0.49912
0.50180
0.5051
0.51344
0.52274
0.53342
0.54567
0.55957
0.57514
0.59233
0.61099
0,63090
0.80133
0.90606
0,98583
1.07320
1.07705
1.18048
0.49850
0,49851
0.49851
0.49854
0.49858
0.49885
0.49917
0.50189
0.50549
0.51354
0.52283
0.53355
0.54583
0.55972
0.57530
0.59251
0.61119
0.63111
0.80156
0.90630
0.98611
1.07350
1.07736
1.18082
Modified
Integral Equation, A-I:
(ii) Fox-Coodwin
0.49850
0.498f0
0.49850
0.49853
0.49856
0.49882
0.49914
0.50184
0.50545
0.51349
0.52279
0.5335C
0.54576
0.55966
0.57523
0.59243
0.61110
0.63101
0.80144
0.90618
0,98599
1,07336
1.07725
1.18070
Table 1.
1lipse, tanh
- 0.5, w-,dra t
0.5
VIRFUAL MASS
....
. ..
Kh
0.00001
0.00005
0.0001
0.0005
0.001
0.005
0.01
0.05
o.l
0,2
0.3
0.4
0.5
r.6
o.7
0.8
0.9
1.0
2.5
5.0
10.0
Table 2.
eT
ModiT
f-l a - -
0.22432
0.22432
0.22431
0.224k)
O.22423
0.22401
0.22368
0.22237
0.22094
0.21964
0.21914
0.21909
0.21950
o.22026
0.12135
0.22279
0.22460
0.22681
0.32557
0.45043
0.56,60
0.22431
0.2243o
0.22430
0.22431)
0.22421
0.22396
0.22)62
0.222 v
0.22786
0.21953
0.2191
0.21900
0.21939
0.22014
0.22123
0,72268
0.22449
0.22671
0.12548
o.45084
0.56149
References
BAI, K.J.,1977. The added mass
of twodimensional cylinders heaving
in water of
finite depth. J. Fluid Mech.,
81, 85-105.
FOX, L. and GOODWIN, E.T., 1953.
Numerical
solution of non-singuler linear
integral
equations. Phil. Trans. Roy. Soc.,
A245,
501-5 34.
JEFPREsS,H. and JEFFREYS,B.S.,
1946.
Methods of mathematical physics,
JOHNe,
9, C50o. the motion of 59.O83.
floating
bodies, II.
CoMm, Pntre and Appl. Maths., 3,
45-101.
JONES, D.S., 1974. Integral equations
for
the exterior acotstic problem.
Q. Jl. Mech.
appl. Math., 27, 129-142.
OCILVIE, T.F., 1976. Private
communication.
RHODES-ROBINSON, P.,F., 1970.
On the shortwave asymptotic motion due to
a cylinder
heaving on water of finite
depth. Proc.
Cambridge Phil. Soc., 67, 423-468.
SAYER. P, and USELL, F., 1976.
On the virtual
mass, at long wavelengths, of
a half-inuersed
circular cylinder heaving on
water of finite
depth. 11th. Symposium on Naval
Hydrodlynamics,
University College London.
THORNE, R.C., 1953. Multipole
expansions in
the theory of aurface waves,
Proc. Cambridge
Phil. Soc., 49, 707-716.
URSELL, F., 1953. Short surface
wave- doe to
an oscillating immersed body.
lroc. Roy. Sc.,
A220, 90-103.
URSELL, F., 1961. The transmission
of surface
waves under surface obstacles.
Proc. Cambridge
Phil. Soc., 57, 638-668.
YU, Y.S. and IIRSELL, F.,
1961. Surface waves
generated by an oscillating
nired
body.
J. Fluid Mech., 11, 529-551.
YEUING, R.W., 191i. A hybrid
integral-equat.on
method for time-harmonic free-surfact.
flow.
lat. International Conference
on Ne'rirco
Ship Hydrodyiu6mics,Caithersburg,
Maryland.
184
Abstract
This report describes a theoretical model for the hydrodynamics of flexible planing surfaces. Linearized potentialflow theory is used to obtain the hydrodynamic effects, and
(hesurface itself is replaced by a finite-element representation
of the pressure distribution. The flexaral behavior of the
planing surface is computed with sinall-deilection beam
theory, and the plate is discretized into structural finite
elemcnts. A series of numericrl experiments shows that a
sufficient degreeof cmvergence is achieved with forty
elements, lhe resullant pressure distributions reproduce the
le:rding-edge square-root singularity - which is well known
in airfoil thery. The rigidity of the plat is foundto have
a major effect on all the reilts: thelilt and drag are
reduced, with this influence being stronger at higher speeds,
Tapering towards the trailing edge has a similar effect. The
pressure disi, butions show thit there is a possibility of
negative pr,_iureregions under the deflected hlape of the
surface - indicating that cavitation mightoccur in some
conditions. Curves of the free-surface elevation are presened,
anti inaddition, the results ofapplying a pressure tn the
plate are described,
1. Introduction
Background
the work descrihed
inthis paperwas prrmpted ty air
intetet
in theapplicion of flexible planing
surfaces
to
Advaned arne craft.
iOne such po igle us.isto support
the weightofit vehc e o thiree
or more
s
itably
splia
ed
theweight
. ifatre etice in theree
ir nreuiblystracit
surfaces. A careful choice oif the degreeoif tlexibility
arid appropriate damping - wouldproducea much smoother
title in toughwater than is given try a rigid
shape.
Anotherappolication
offile
con-ept
isit,theseals
o1fi
it an SlS fifted
stal-eclFet ship hSfl A typic:d laytrut
Figure I
with sitlcwalls and txuw anI sterfn sealsis sfown ill
tire train desigin feature of tie sealsis that they call deflect
excessive hs
it%a sea state without
in order to firvit ni.ivel
tire craft
if pressure from tire air cuhion which supports
willuiot he given here, but tie
ilre details ofthe seatldesign
ihii hinbred.
trailer is referred in teart to other works tit
Ore i thephenomena of great interest - Aswell as tire
trenerated force, - istire wavesyster dexehrrd uniter tire
then
(fllireritp
vehicle If tile sidewallnare not
where a
air call ruccueinthrise u i rions
if trrhir.)
venting
of the
attitude
in
a
Iower
[hi
results
exitst.
wave trough
to the
hydrtrdyrnAi drag title
ShESwith antattendianit rie ill
buddur i water Againstti'
seals
approachfor
a finite-elemnt
18 1 developed
Doe
plates of artitrary siape, including tire effects of
handlinig
graviy waves, The methtod Alsotallowed the predtiion of
optinn Irfrris which trircided with thoseof C'umetbatch.
With regard ti thre-dinwnisitmal plning. themait
'Permanent address
Skchil ot Mechanical anti Industrtal Irgineerig
Univerlty
of Ne w SouthWales, Sydney, Australia
185
STERNd
SEAL
SUPPLY
CUSHION
SUPPLY
BOWSEAL
SUPPLY
OWR EIIFNT
1 FRWASRE
SEINFIIT
PRESSURE
RAND
UNOEFI
ECTED
STERN
EEAL
SHAPE
POSSIBLE
CUSHION
VENTING
jf
UNDEFLECTED
BOW
SEAL
SHAPE
FREE
SURFACE
PROFILE
4. AFTHALF-TRIANGULAR
PRESSURE
ELEMENT
5. COMPLETE
TRIANGULAR
PRESSURE
ELEMENT
M-
TYPICAL
PRESSURE
3. FORWARD
HALF-TRIANGULAR
PRE
SSURE
ELECMENT
P.
TRAILING
y\
FREE
SURFACE
.0
EDGE
xi
2-
1).l~
NOTES:1.PRESSURE
91H4A41N
ACTSDOWNONWATER,
AND
UPWARDS
ONSEAL.SEALPRESSURE.
p., NOTSHOWN.
2. CUSHION
ANDBOENO
PRESSURES
UStUALLY
EQUAL:
P. I Pb
Figure 4 - Theoretical Model of Sealand Cushion
Ffesnt
orkthe
Theorv
aeoml
Irmast
eieeiaii
rmssctnu
eli
h elce
is siiwn in Figure 1. "Tis diagramtAlsodefines tire
convention for the nmomentM, the shear foirce V. and
loading p. Therelationts between these quantities uni
the deflectioni w are givert by stan-lard beant theory (we
91W
ong1i,1
n
msek
d
(II
p
'Is
~sign
nalticther
Thetomof
I todevlopin
hiituy
fir the hynirdynarmcnu tifexible pilanringsurfan-es.whtich
coualdthen be atipliest to an SES. One 4imphifyting assumpliron ti lheutiized is t1%4tof' two-stimensi~iurtly.
It is, of course, well knowrt tliat the flow under the
craft u%aiually thrte-dimnsional, and it would therefore
went approriate to use the three-ditnensionat tinite-elentent
planing technique oif Doctors 137). Thin could be continedh
with the theory for the wanesgreneratedby a preaure patch
(itn oeuter tur inodel tte cushion) reported by filuing and
Worq 1381, Ilausullagi and Vin isettirre 1391 and Dcis140)1
ISO
V
aid
1M
x0
2
M.- T)- t
dn
(3)
where
3-4
,,,
(4)
M
M =;pX2.
= I ,
(5)
= i-
(10)
w, = w +
+
i
and
(6)
D = D,(11+ atl,
(9)
M = 4p a (r + t)/y,
wa
55
or) In I + a
Pj
a - (I
ui
iy ..
2awi+L
I I
[n2
Di + I/Di and
n
in w:iich of
station spacing. For convenience, the i index on a will he
dropped.
+ 21l - or)(m I +
w=w
(I -i~nl
bill
(24 I)
4p,a["
tail.
(7)
a
and
8 pa
W;
+O
t+
ij) 0-lal
luill
+a 1-1
pa(aI-
p
2pja2
(14)
I5
.
"
+ n)
(8)3La
ll + 3a.)Inkl to )
nt
06)
4
,
and
w " w
- 2a w'.
i + Ia
0Zi
lIS?
(j
4
,i - L i+I- I20
53sIaoz
(12)
-a
l3a4
L.
)i
(I + ,)
a
L
5
2
bnll +(
(l71
0.12
0n0
10
0__________
0,012 -n1
0.120a.
0.06-
0.2
0.4
0.0
0.8
1.0
0.2
C. 0.006
Ca .0
0,4
0.6
0.0
1.0
Figure 5
'---0030
____
'.o
n.tO
0.012
:025 0.
7 0
'a.131
-2-
0.020-
10
_5
0004
0.2
0.00--.
0.4
0.6
1.0
0.0
.-
0.2
0.4
t ie
4 p" , [F
W,
"!
--.-+-In
.
w:
4
Du L
IM3
'
the three
riiltereiii aviations bretween thresource elenient id the Iteld
poinlat- as for the loaward halt-triaiular eenct I lie
resuls or lite first case,namnelII j, can he otainecd
tarevctiyfromn those of lite horward elemnent. fil onraly
iflernce o-:Luis Ili colllna
41w
fit oarren arin in 1-,1 (10)
whiicha
shoaaid raw riot
cvrarider
Isurlaice,
r
sthat Val,. It
---
24)
1
Ca
t
a-
ri
( 20)1
I aIra
t wiarhirled spralcly. ra.I
varloaading. shear foca lii)
nd~ rllera
p4
1.0
+1]40
Itli third Case -id thaI [h tird Ioint living located
behind file source element II j- I I Ila%already beein
treated, by 1:11s(INt and lit))
t ie case )I i
ever. I lie equatirair
are reIKectrvety
0.0
Ill 14*
I~
AtHl-iaghEant.We
0.6
I f 3a - 3n + a') In .Ia
lea
(27),
and similarly
4
8pa
,
a3
Inl+
wvi= w, * i -2--aw
(28)
=-
Results
Pg/Pj
where
X, = xsk0
and
k0
w
w --
Pit 4
(2
(29)
(30)
gIC,
,f
asX,
X.
0
(35)
f- 1
Cost(, f as X
(3h)
(37)
- -a
as
+t'sI
-
sgnlht),)
I[/wnhnfo!
+ It-
:mlx) - I
dx,
ll-a2,:s(htI
ao
2-
0.4
_.k._
2.2
--
-.
Figure 6 - Hydrodynamic Influence Coefficients
Ia) Semni-Infinite presture Bands
0.4
_____---a
", it.
0.2
Pi 0
0.
-0.2
.,
--
0.025
000l %
-o~N
-o
-4
-8
oeTcet
Figure 6 - idoynmcIflec- Hyrodnandk~nluene
CeftintsFigure
bForward Half-Triangular Eleny.-nt
sg Id
XIf(
6 -Hydrodynaml
Innlunce Coefficients
opeeTinua
wd lmn
hInt(X)
X
Its asylltotki
The results for the waveeleVation, ii,.
are
form,+
4,1
IIIA 4+ it - A, I(2 inB ( A
%gnlh,,
If AXo)4 Itt- An)(2
where
where
as X,,
2at,
2a)
q. (4104
to Li.(431:
p~p
-.
(41t
2hi,
(A'
2 1
Aft Hit-Trimsptlar
Ele wit . In a %inilar
maniner to the
Previous Lase one can consatructthis rlment usins a (positive)
aft srm-lintlnile
hand sltting AtA 0, and then subtracting
an integrated seriesol' bond, The final result is
~ak,
441
-IeeIe
(09)
ak- (sinh
(A4I- sin1 1)- 2 co, (Xo
(xl + 2 a) k0
if
I-
l)
4 lurtl tak,
skl
(2sin (hA.)Au
aj4X
Igo
X,
(srr1
12u
A)-h
-(
t1(a
45)
Results
the SealDoflection
wi =
h for I C i C n.
(49)
The kinks visible in the curves for ak0 = 0.2 are due
to the fact that points were computed only for vaues of x
which were an integral multiple of the element length 2a.
They could have been avoided by using a closer spac;ng of
points.
IV. Combining h
-;
i=2
p ,
(50)
n
D =
D
h
Pb"2
+
Elements
wi +
W2- Z1 W
=2 P (i +
(51)
I -I - I - wi -1l
Kinematic Condition
We now apply the results of the previous two sections
to that of a flexible planing seal travelling ahead of a cushion
pressure on the free surface. The configuration is shown in
Figure 4. 'the cushion nressure, p., and the seat pressure,
p, are normally identica;, but will be kept separate in the
problem foimulation.
The solution to the problem is obtained by summing
the wave profiles iesulling from all the pressure elements,
and equating this to the profile of the deformed plate:
(rot )-;iIfi(
j iI
-pb(j-'
pgth +z(z
i
~-expressed
)
fori= I to n.
and
xEP
(47)
Dw
'
C
pgsA,
"
P ii(" +
Pj
P $i
Pt
tg
448
14asv~
itlak
r ....aku
Pv Pb
4 in'Itki , n
-psinlx
a
Y-'
rin
A55)
and
Elevalion
(54)
A, cos(xku) + Assin(xk,).
1531
-ottPil
52)
i -2
Frw-u e
M=
pg A,
2P
-4
p %
k.)
(5li
xt
saku )
aku
4'.2t"
5
aka
p, Cosi
(57t
1.00----
1.00-0.96-
FG.0D-0.001
0.95-
-------
0.90-
-----
---------
F 6. 00001
0.90-
1.D-0.0
0wit
0.050-
'
..
*wP1.~0
075-
0.790.701
0.70
0.
05
F6D
1.
10
1,D
2.0
20
1.-LOG'
5
2.6
.
0.1
---
.1
0.5
-----
1.5
2.0
2.b
LOG2,,
of the Numerical Scheme
Figure 7 - Convergence
(c) Tip Defletioan
or ItheNumerical Scheme
Figure 7 - Convergence
ia) Center of Pressure
it)
2.0-
2.0
205
f 0,'
CF
iure
onegnc
2.
F-1.0-
of th Nue-
,D .JO
0.VI
Scem
ilicitilonnies rlifidity*
.0
02
0
Fir 7
D*
lilt "WIle'icient
C'
Dlii cereicteirt
(it
)4nQ
1.
Scem
I, Illustralvd Ii
thlie ewt work - lte seal%liffies%. 'his
tgiurv Stall thrOiig fill, III thee CUtirpiCs. lte seil
andt
tife
e ctish
uil
Ih
pomevi, ICiit littitickne-s
seal p;iie se will IseShowin later.
( S9t
'/V
-[
of th Nuei-
teNmeia
I.
0.4.-06
Co-re
an
Du0.001
Fut1asi
'00
03
0.5 1.
Figur 7 - Co-rec
p *
(59
talinig cdge.
]c
it(tl1
PC" q"2
-
192
---------.-
D - 0.05
'--
------...............................--
n-001
----
--
- -
- -
0.
x0.750
0.7-~
0.7-
07 0.00
--- 10
O.b
0.5 -
n-so6
3
F
F
Figure9 - Variation of Seal Taper
(a) Center of Pressure
.i
/"'-
10-
2.0
n- U
- 0.06
1.0-
SO
n-. 00
-
2 5.
-2C.
CI.5
C,150.6-
r-0
1.-
O 0
1
-------------------0.01
--
0.5
5
4
3
F
Figure 8 - Variation of Seal Stiffness
(b) Uft Coefficint
2
" "......
----
0.761
1-
0.4
3
F
2
--
2,0-
,.-
r-0-
0 oo
...
:::-_.- -----------4
F3
--
0,001
0
.,r
F
3
- 0..
-- ---------...
""
1.0
--------.....--...
-.
......
D'Ove
t
0I
0.41
a
/.01o4
0 . 9o
&4
. . ..
"
I.
8
Figum
025
0,
/0
'O
"
1
FI
13
F
Figum
- Variation of SMaTaper
(d) Tip Delctlon
193
0.
02 0
1.
040
in 0
0.2
0.
,4
.0n r
0.
.6
1.2
0
1.56
0.4.
0.
0.6
1.2-
.4
06
0.
0.
///f
naN
060-1
Presuro istibutonsFigures
Pragar. Dileibetlonaaltered
1,e nuiw turn Ii) F~igunre1ll whnin four part, Ilistrate
the nivid effects of the varii piaratreters irrtronhmevl above.
For the prptne it ilust rationi, thle Irrsure ban cern irade
dirnimonlevn usng the lift vif*the seal. L..and it, lenth, V
A reatively Ilenirle seal in considered tin I ifure I that.
An a renult, negative tirenvures are evident ove r nirom regionis
of the aureface At low Frinide numbtters.tire wavelenglth itt
the prrnuorr nncillationx in practically the natie as the %tattdard result frot linear theory. tits the seal is emnrrualy
caintoratinglvio the natural water iwanegeeratd by thec
forward tirrtiiin irf the seal. wiron deflection in obviusly
lens.
Al higher speed%,ncr) little lift is carried by the Af
pornnit of lire nurfoce. Altar. region of negative vir very
low relative prsares %tilloccur. Of nole, is the fact that
the wavelength iir the otacillatimiais inaci johtialer fiie the
higher speedy. ti's cotttradicts aimple uncirrntrained water
%ave theoaryand porint%
up the influence of the %tiffts
(theit '111411)iofthe Wal.
0.
ade
tr
p'/pg On16
P,
p'r,/nga
0n11i
~~0
0.3--
0.3
0.2-
0.
0.0
1.5
X\
I""'
-, "
F-
.__________
,6
I -o.
t/
-.
...
___
r0
n0-
/ oNo,..,,
0.
0.
--
Ile
>..As
0.on
-~01.
to0
-2
-1
10.Bo
1
4-l
6.0
2-
42 --
D1"
- 0106
F 05v-l
-~
-12
'/P*
Oa
:5
n-80
-
-2
-1
2.
-t
x/1
./V
(dl D
WaE Profile
VI.
p.
Conclusicls
Prosnt Work
The principal object of the researh described in this
paper w, tii inireoluce the eftect ot cornplio-' e of the body
into planing theory. Flexibitity is seirt l hj, a proftound
effect on allttie quantities of interest. in brief, increasing
the flexibility results in a toss of lift and a reituction iii
Irag. Atso, the cenici tifpresure moves lorward. Incvreasing
the Froude number makes the inorirtance if compliance
greater.
leading to
I he last figure to be presented, Figure I Id). detnonstrates tilepowerful effect of applying a pressure it the
cushit
,1 the teal. In this case, sie seal dellect,
dawnuws, .
inal is, a ieialive deflection iscurs). Ihis4 efre.t
is due to the pressure acting down on the seal. Very large
downstream waves are also generated. A cirrisr nin ofiware
heights with the previous parts of Figure I I highlights this
feature.
Inctitentally, it thould be pinnted rut here that the
value of the lending pressure element, pi. was cbson
p ovide a continuous presure distnhbuion in Figures Il(d)
Future Work
An obvious improvement in the ruleot convergence ot
the numercil stherue coiuld he achieved through use of a
st ctil pressure elensent at the leading edge ti the planing
surface
Ihis element would need ti pisa
the situare--rt
singularity referred to before. Fsaminati,i of sorneof the
ae,
.- "ufc
.,adLioe
1.Wlisn
Encyclopedia of Physics, ''o. 9. Fleid Dynamics 3. ed. by
S. Flitifc, Spcinger-Verlag, Berlin, pp. 446-.815 (1960).
0
Acknowledgemenits
'fie research described in this paper was perfPrmed
while the writer was loicatedl in the Aviation and Surface
Effects Deparl cot I ASli) sof the David W. Taylor N-it
atring
Ship ResearchiandI Develospment('enter (1 1YINSRI),
leave from the University oif NeswSothtl Walesin Sydney,
Australcs.15.
sit
[lie writer wouldl also, like is, thantk D~r P R. Zirdat sil
tile Nuntiticl Structutral Mclhit cs Blranichof L)IINStRl)
tsr help1 that lie gase.
I8 Doctor%,I A1, K i-Isssitatlsl I laing stiefiiiihi Fial Prnstir- Flertnin(, Prsic PIlth Atslt.i.-ant
tislilericec
llvs uhclsvitli E'litis Ms-cl.voscv,Viol. 2.
1,1t' O 4,148 (1974)
Rohietr
i ,o s a (;ltiirs
1 Srvlettskss.IN. -0iv thst ...
,auk ShSR tisel Mkt. I steil
Deep Waler, Its. Akad
Nati, Vist (i3t till 8178His (i I t11illit I'1131
2Sretc
ii.
I N
"Or
t
thf
SS'KkislNasi
leo
ittel.I e~iss
pspiit..1.2l,1
titif
140
~ti
3Sediss, I I ,
'tunale
(I11fi
1 17)
4
Aiss
'021Ihls
.. It
lss
i fithrSalc'
Iloatiug at tHier Yteetls." Scltil istec(iltk ViI. 4, Ni
I'(4),
ssslttteiiisl
1t ,.
)SJ7)
pi
.*r'.o,
toissl
fil
Ac-Gisstiiaui.i hut M....Iisiswlf;
PWstIsle,,
... nl.- , Mu
is
-,
Mantis
Gidtsing .. , the
, 1-idy- Kisitteisvtol
I ceitlrili
I1I
_______
Il '5
hlIts-Pane Nbhli
a thillind
So
Nisk. Vt
I lie.., ioif
Narsra
*NtoalAiif
Rtesr
(NAtAl.
li1.1
isliilslte
(otlifc
1
'e
sI
ci~t
tt
((1158K1
oI
lt. I. MI. /.it, Plann ',I a hi- I ilsist-Rait,
m~eli
be' 1,itihil
I nse"t
lnitficeIstie7ii~r
Iinicsteir M.ithetvia.t!" If pti 1197 1
22
the II Irs
Ikit Ships~al
/~li
11M.imoii It . lIil
114-st
7
Mari. t 1
(Isi1.
P'il I Vol
10 s. ('t
21
lo I lI~
o' )
~i%
5fi
Rts'irt,i VsiI
aIthjling2
Vis
dile
41 pp
lob
"i.
lii.
Wtit,
sniitteh
('14141
lrniienidi.
''i.I Slu
isj Watieflirfi.'
eltiiisiINtlIMI
I lrauis]tln i of
.itiiil
s
soi
n'
?w-Aspert Ratio .4pproxi27. Marur. It.. "hlig'
matio iof Pbr "tg Sw'fa'-s.'" hifL~technik. Vol. 14. No.
72, pp. 57-64 fin Englishlll9ti7)
DISCUSSIONS
of three papers
Invited Discussion
H. Maruo
Yokohama National University
Dr. Yeung and Mr. Bouger have developed a
method that looks quite novel as a treatment of
steady free-surface flow problems. The special
feature of this method is first, the employment of the ordinary source potential inplace
of theconventional Havelock source, and second,
the division of the fluid domain into an inn,!r
region and outer regions. The first point is
advantageous inmaking the computation of the
kernel much simpler but the evaluation of th"
second derivative of the velocity potential on
the undisturbed free surfac necessitates some
special technique such as t:-employment of the
spline function. ",ne
division of the fluid
domain in inner and outer regions at proper
radiation boundary isconvenient especially for
the case of an Irregular bottom. This Idea
reminds us of the method acopted by Yamamoto in
1975. Inthe outer regions the potential is
expressed by an asymptotic expansion. If the
radiation bundaries are taken far enough from
the body and from the irregularity of the
bottom, only a few teres are necessary; otherwise the local irregularity will cause some
complication. This situation is particularly
important in the case that the bottom isnot
flat. Unfortunately, only the case of a flat
bottom Isgiven as a numerical example. Further
calculation for irregular bottoms isexpected.
The authors have also referred to the NeumannKelvin problem and have expressed optimism
about the resolution of theline-integral controversy. However, thediscusser is rather
skeptical about this. A numerical method of
this kind can be safely applied to a problem for
which the existence of a unique solution has been
proved. In the case of fully submerged bodies,
this might be allright, but there isno evidence
of existence of a stable unique solution In the
case of surface-piercing ites, because the
linearized free-surface condition is quite artificial. An interesting result obtained by
Bessho concerning a vertical plate Inserted In
a uniform flow with free surface showed a nonuniqueness of the solution of the boundary-value
problem with exact boundary condition on the
pte and linearized free-surface condition.
9esho has proved that the integral equation has
aigensolutions that satisfy the homogeneous
Discussion
by-wa-ng-June Bai
of paper by Ronald W. Yeung and Yann C. Bouger
The method presented here was applied previous by Yeung to two-dimensional time-harmonic
free-surface flow problems and the results were
presented at the First International Conference
on Numerical Ship Hydrodynamics in 1975. The
same method was also applied to the same timeharmonic problem by Kim (1976).
For solving the uniform steady-flow problam considered here and the time-harmonic problemsconsidered in Yeung (1975) and Kim (1976),
I personally do not see any advantage In using
the present hybrid integral-equation technique
over the conventional Green-function method
(distributing singularities on ony the body
boundary). It sees to me that the computation
of the coupling terms, which result from matching the elgenfunction representations (i.e.,
cosh m (y+h), cos m (y+h) and the fundamental
singulgrity distribItions (i.e.,log r) along
the artificial juncture boundaries, is as conplicated as the computations involved in the
conventional Green-function method for the
finite constant depth case. This fact is obvious when the Green function (in the conventionalGreen-function rxethod
is expressed in
series form.
In the earlier work of Yeung (1973) and
Bai & Yeung (1974) the fundamental singularity
method is used without matching of the eigenfunctions. In this case there is a computational
advantage over the conventional Green-function
method since the kernel is much simpler. However,
in this approach, the computational boundary
domain is increased as a trade-off since the
fundamental singularities (logr) are distributed
along the entire (closed) boundary.
I would appreciate it if the authors could
express their view on the advantages (or disadvantages) of , , present method over the
conventional Green-function method for the case
of constant water depth. It would also be
useful for the authors to discuss the improvement of the Yeung (1975) and Kim (1976) methods
over the ear'ier work by Yeung (1973) and Bai
i Yeung (1974) for the general cases including
the case of variable depth.
Py second coment concerns the authors' use
of spline-function approximations along the free
surface in the present work. It seems to be
necessary in the present formulation to use the
spline functions due to the presence of the
second derivative *xx in the free surface Intogral. However, the seco
derivative
in the
integra1 along the free surface In Equallon
T3.3 can e eliminated by Integratn by parts,
resulting in a slightly different fom which
involves only the first derivative as the highestorder derivative. Then, in the new formulation,
It Is not necessary to use complicated (and more
sophisticated) splin functions. From W own
experience. it is mch easier to use simple
piecewlse polynomial functions which permit discontinuities in the first derivative, #x, at
intersecting points of two adjacent segments.
I
would 1iLe to ask the authors if there Is any
other significance or advantage in using the
splice functions.
Finally I would like to point out that the
In response to Dr. Bal's first query, related to the computational difference between
our hybrid method and a traditional Greenfunction approach, I want to point out that the
coupling terms that Dr. Bai refers to, when
computed from the expressions in closed form
that we obtained, are about the 'aie order of
complication as the Green funrtion itself. However, in applying the Greon-function technique,
one is required not merely to be able to compute
the Green function, but also 'o be able to integrate such a function over a segment of finitesize. This requirement of the distribution of
singularities is well known to ThoseappTying
the singularity methoo. Generally, integration
of a Green function over a segment cennot be
carried out in closed form;thus it must be
irtegrated numerically. lherefore, our ability
to obtain closed-form solutions of the integrals
associated with the radiation boundaries means
that we are one step ahead of the Green-function
method. Such closed-form solutions are central
to the success of uur 'cheme, a fact that was
pointed out previously in Yeung (1975) and in
my student Y. H. KIm's thesis (1976). As far as
computation time is concerned, the hybrid method
of Yeung (1975) was about two to three times
more efficient than the original formulation of
Yeung (1973). A recent work due to Harten and
Efrony (1977, J of Comp. Physics) indicates
that by block structuring the matrix a further
improvement by a factor of 10 is possible. Thu
it appears that our hybrid formulation has a
tremendous amount of potential in computation
economy.
Eai, K.J.,1972, "A variational method in potentialflows with a free surface," Ph.D.
Thesis, Dept. of Naval Arch., University of
Calif., Berkeley
Kim, Y.H., 1976, "Hydrodynamics of cylinders in
water of arbitrary varying depth," Mastci's
Thesis, Dept. of Ocean Engineering, MIT,
Cambridge Mass. 67 pages.
Author's
21R
by R.W.Yeung
to discussions by H. Maruo and Kwang June Bal
First, concerning Professor Maruo's comment
on the possibility of complications in applying
the method to the case of non-flat bottom, I
would like to point out that similar techniques
were used very successfully for such a case in
my previous work (Yeung, 1975) for the timeharmonic problems. Although we give no results
in the present paper, we anticipate no difficulties in that direction. A distinct advantage
of our hybrid method is that the choice of the
truncation boundary can be quite arbitrary and
in fact can be as close to the body or the Irregularities of the bottom as one desires. Any
localdisturbances can be absorbed automatically
in the series expansion,
With respect to the point Loncerning comparison of computation time of the traditional
Green-function method with our method, it appears
that there exists no published computational
data using the finite-depth Green funztion for
two-dimensional steady flow; hence such a comparison is not possible. Perhaps this is an
indication of the fact that the Green-function
approach has not been too attractive computationally,
The second point concern% the treatment of
the second derivative term of # on the free
surface. We want to emphssize that a spline
approximation is not absuiutely necessary, although it was found to be most helpful. We
could have used piecewise polynomial approximation between successive grid points on the
free surface and this will result in the sam
type of Interals as in (3.14). Inasmuch as
the usage of the spline functions hardly cowplicated the calculations, that one obtains
smooth first and second-derivatives at every
grid point, and that very fewpoints are necesary to rtpresent * in
a J. The high
quality of the differentiated curve as shown
confirms the usefulness of the techniqie. The
application of spline functions to finiteelament methods such as Dr. al's, however,
rC,
Author's Reply
by . ayer and F. Ursell
to discussions by H. Maruo and
T. Francis Ogilvie
We are grateful to Professor Marjo and to
Professor Ogilvie for their illuminating comments. In addition, Professor Maruo ha.;put
some searching questions to which we offer the
following replies:
(1) We have riotsucceeded in applying conformal
mapping to the finite-depth problem.
(ii) We should be glad to know more about
Ohmatsu's work on transient problems. The work
done on transient problems at Manchester has
proceeded along different lines and does not
involve the complete determination of the water
mot )n. (See S.J. Maskell and F. Ursell, J.
Fluid Mech. 44, 1970, 303-313).
(iII) We agree that our device of using the
transposed kernel must be equivalent to the
source method used earlier by Besshu, and that
theadvantages of that method (provided that
only the total force is required) deserve further
study. The irregular frequencies remain the
same when the kernel of an integral equation is
replaced by the transposed kernel. Thus they
are the same for the Green-function method as for
the source method.
(iv) The eigenfrequencies of the interior
Dirichlet problem are in general distinct from
the eigenfrequencies of the interior Neumann
problem. We find it difficult to attribute a
physical meaning to the interior Dirichlet problem in the present calcuiations. Indeed, if
the source potential is modified, then the corresponding irregular frequencies are also modified and are no longer related to the interior
Dirichlet problem.
Author's Reply
by L.J.octors
to discussion by H. Mauro
The author would like to thank Professor
Maruo for his interesting comuents.
The first point concerns the unknown length
of the wetted surface. As pointed out, the
position of the leading edge cannot be determined beforehand--assuming, of course, that
the height of the leading edge Is specified. The
only practical method of skirting this difficulty
2I
Abstract
Th
-Its of an investigationof the applicatio. of ... -4cally-genetated boundary-fitted
curviline- coordinate systems in the finitedifference solution of the time-dependent, twodimensional Navier-Stokes equations for the
laminar viscous flow about hydrofoLls moving
either in a free surface or subserged at a
finite depth in a fluid of infinite or fi-Lite
depth are presented. The hydrofoil may te of
arbitrary shape, and its motion may include
pitcbing oscillation or oscillation normal or
parallel to the plane of the undisturbed free
surface as well as translation parallel to
this plane. A computer code has been developed that is capable of predicting the flow
field, pressure distrt -',ns, and force
coefficients for this contiguration at low
Reynolds numbers. The finite-difference solution is implicit in time so that all the
difference equations are solved simultaneously
by iteration at each time step.
I. Introduction
This report presents the results of an inveatigation of the application of numericallygenerated boundary-fitted curvilinear coordinate system in the finite-difference solution
of the time-dependent, two-dimensional NavierStokes equations for the laminar viscous flow
about hydrofoils moving either in a free surface or in a fluid of finite or infinite depth.
The hydrofoil may be of arbitrary shape, and
its motion may include pitching oscillation or
oscillation normal or parallel to the plane of
the undisturbed free surface a well as translation parallel to this plane. A computer
code has been developed that is capable of
2V2
about two-dimensional airfoils has been reported by Thames, et. al. 131 and by Hodge [4].
The latter reference uses the pressurevelocity formulation used in the present work,
(C)
(lb)
with Dirichlet boundary conditions, one coordinate being specified to be equal to a constant
on the body and equal to another constant on the
outer boundary, with the other coordinate varyIng monotonically over the same range around
both the body and the outer boundary.
iY
.-
2
J Ix1 1(tPti)+ x Q(C,")]
--
ay
where
a -
- 20yen + yyn
2[y
2
P(Cn) + yQ(Cr)]
xZ + y y
C'n
(2b)
2
C
+
11
(2a0
- K
xny
x
Th system described by Eq. (2ci a quanlinear elliptic system for the coordinate functions x(CEn) and y(&,n) in the transformed
plane. This set is considerably mora complcx
than the linear system specified by Eq. (1),
but the boundary conditions are specified on
straight boundaries, and the coordinate spacing
in the transformed plane is uniform.
203
af
- Ft)"
W
ap(E
I .f !X _
+ If
foil solutions, is to choose P and Q as exponential terms, so that the coordinates are
generated as the solutions of
tax +. E7y
(=f ).
.
D(x.y.f) / 3..I
,ii
a(.,.t
= a((.n.t)
xy
a.
. an an )( .n!
x..
fx
(4)
at
si-
exp(-cik 7 E
exp(-d
I)-
.(C
?( ,r)
+ (n -
All derivatives are expressed in the transformed variables (C,n); thus eliminating the
need for interpolation between points in the
physical plane. The movement of the physical
(t time
plane grid points is accounted
ax for by the
).
= bj sgn(E J1
(3a)
and
.
4
Ai an(n
+yy
I b
J-l
sh.tn
(-
n )es(-cjln
ni)
(3b)
nj)
(n - n)
Q(
,n)
+ (u2)
+ (uV)
t
-
(a)
+ uyy)/R -
p + (u
v t + (uv) n + (Vn)y
-
3 2
2uv
u-
Vet
where I - -- and 7 -
(Sb)
1/12
py + (vX + v yy)/
..
1
D
t
(50
V
-
204
+ [yn(uv)
(X
+(p
.((
)/RJ iv
+ ov+
n
Ac 2
V(
'n
YvM
i/F
(7b)
the velocity components of the hydrofoil surface at (x,y.t) relative to the coordinate
system translating with the hydrofoil. (These
values are zero if the hydrofoil is not
oscillating.)
- 2x u
- (v
xNut )(y
y Cv
- x v )2 - j2D t
-tDtx
D x,)J
-D nyt)J -
Xt D
Y(Dx
Yn
nx(D
(70)
where
+
2
+ I
i vyu2 +i (uy +x
)nl - (p
)
p,)n
- yun +
D -(ynut
(6b
(6b)
and
EnV)IJ
-xv
-
j o(nI,
(7d)
conditions apply on the remote boundary strictly only until surface diavesreach it. At 1oe
to damp the wmvis beore the remote boundary
10 chords distant is reached.)
The time derivatives have also ben tiansformed In these equortens. Thus, time derivatives in Eq. (/) are taken with C and n fixed,
while those in Eq. (5) were taken with x and y
fixed. This transformation of time derivatives
allows the computation to be done on a fined
grid in the transformed plane even though the
physical grid is in motion due to the free surface and hydrofoil moveme
it -
yuQ/J - Yt(XU
u - uB(C,t), v - vB(,t)
- x,1u/)/S
S( 2) C
yn(p - po)J
(a)
x NO)C/J
2
+ (yipC - ytPnl/J
=lNu,,-
V Cy /J u_
1nnqn+ (nO + Jxny )v0
2 C(- Jy)u
2$u,, + YunnC
+ Ou
++ [x(uv)
((nB - JXnyn)u
(u2)JC/J
12
+ ru )/tJ +
(Ob)
n(p - po
(7s)
flow):
(c) On the remote boundary (undisturbed
2
- Vo,v - 0, p - po + (Y. - y)/
.
vt - Xt(oqV
205
stress equations (6) to the curvilinear cootdinate system end solving these two simultaneous equations for u& and v& in terms f u,
the free surface being a line of constant F in
the configuration used. (See the appendix of
[7) for this development.)
(9b)
(k) - KD
F - - 2
with D given by (7d). Here (k) is the teration counter, and K is a proportionality factor
gvnbKC2.j2
K
( Iby
y)At
on the hydrofoil and
by
K -
px -pyd
(lla)
ydC
y C - ycv - x u
( j C
n
n
(1b)
an)/J
)
(12)
CL - F7 coee - Fxsine
(f
wdt
L -(.t,
,) -
px E
Fy - +
Ia
wd
ydo + R
,m.n
tt
)xtd
0.3a)
bo)
xz
(k)
"(k)
lit
Sx C
V.
Fiur
xx
and then
y
vX u 1
IV.
Solution Configuration
Numerical Solution
206
"),c
(
8, C7 . C 2 are added to the basic
transformed plane to provide free surface and
"infinity" boundaries. The two common reentrant boundaries,C and C
creatn
r are created to pro6 a
Typical resultssened
given in Figs. 4-9 for a larmn-Treffts hydrof,01 (Fig. 4) and in Figs. 10-11 for a circular
c) dAerhydrofoil. The airfoil was defined by
37 coordinate points and was located one chord
below the free surface. The field size of the
rh"outer"
coordinate grid was 54 x 30. The
boundary was located 10 chords from the airfoil.
20?
Computer Time
Some of the solutions were generated on the
UNIVAC 1106 single processor and the latest
solutions were generated on the upgraded UNIVAC
1106 dual processor. There are many factors
which determine the computer time required for
a solution, for example, the way the object
program is loaded in the computer code. The
uncontracted coordinate system requires from
3 to 6 minutes to converge depending on the
field size and convergence criteria. Depending
on the type of attraction required, the contracted coordinate system took up to 30 minutes.
The acceleration parameter for pressure was
1.8 and the acceleration parameter for velocity
was 0.8. The constant of 0.1 was used in the
pressure iteration on the body. At Re - 20
and F - 1, the solution took 239 minutes to
generate 600 time steps. The maximum number
of iterations for a time step to converge was
11 at time step 313,
VI.
The only modification necessary to the submerged hydrofoil solution discussed in the
previous section here wea the change of the
configuration in the physical plans to include
inflow end outflow boundaries with a solid
bottom between. In regard to the coordinate
syetem, this change is merely a emtter of
changes in the input to the program, replacing
the sa-circular outer boundary located at a
great distence from the hydrofoil with a
boundary consisting of three segents--inlet,
outlet, snd bottom. The undisturbed flow
boundary conditions used on the seal-circular
outer boundary were then replaced with undisturbed flow on the inlet and outlet sagments.
and free-alip boundary conditions on the solid
bottom. Inviscid boundary conditions were
208
VII.
*
*
+ &yy
Y) P(f)
+ nyy
Both experimental results [101 and tho present numerical solution iiicate that 'he sft
local disturbance moves aft as the Froude
number increases (cf. Fig. 5). The
Reynolds number effect Itithe present results
is even greater, however. The local disturbanes is broader and extends farther aft In the
numerical results at the lower Reynolds number.
At the sast time, the initial rise forward of
the hydrofoil is broader at the higher Reynol(t)
numbir and extende tather forward.
(t2 +
(14a)
(14b)
I - 2 . anti
i.constant on
I -
- 4
C 1 o
C - C 2 - constant
n varying monotoncally from n I to
(n 2
"1
from I to
2 and from 4
2
to 3.
2OW
40
I -
that the
In t'e present application, the control functions, P and Q, are determined from the specified sparing of points on the hydrofoil contour
fid fre surfpoitose on the hydyofoingconour
and free surface, those on the body being concentrated near the free surface and those on
the surface being concentrated near the hydro'oil as in Fig. 30. The details of this deterviosrion of P and Q are given in [ 8]. The
resuin tceation of
oordQaregivninat
nes
resultant concentration of coordinate lines
rear the body and free surface is evident in
Fig. 30.
In the initial
stages oi
this study,
yrfi
hl
anann
the
twosubtended
intersections
the free
This
angle,with
of course,
ose,
of
e
t
anges ibis
oscilhydrofoil
the
When
time.
in
changes
lates, the movement follows the oscillating
the
awen
control functions, P and Q, were taken as sums
of decaying exponentials that caurs attraction
and/or
lines
of coordinate lines to specified
points as used for the submerged hydrofoil.
Some of the results given below were obtained
on coordinate systems using this type of control as will be noted. The new control procedure has the advantage of automating the control and e,iminating the need for judgmental
estimation of the attraction amplitudes and
decay factors necessary to achieve a desired
111.
lie
o
degre
cnceraton.Section
aYF -
26x 0
+ Yxnn + aPx
28yq
E "y'
cy + + Pp+
+ fQx - 0
YQy
Results of the numerical solution are presented for a circular cylinder hydrofoil In two
flow configurations:
(15.)
(15b)
except by the free surface, with no disturbance remote from tha hydrofoil.
Is the translational case (a) the acceleration is linear, with the Reynolds and Proude
numbers given by
P - 2t
r - 20t
tree numbers being based on the cylinder die- toe and current velocity.
(c)
free surface.
For the plunging ce* the motion of the byr.foll is einusoidel with the elevation of the
cylinder axis relative to the plane of the
intiUlly .ndisturbed free surface given by
y - A sin(--) where A nd P are the amplitude
and period,
The
This point dietribution on the body and remote boundary wes taken ccording to equiangular epacing over the body and remote
cue
ooA
210
20y and P -
2y,
Conclusion
References
1. Thompson, J. F.. Thames, F. C., and Mastin,
C. W., "Automatic Numerical Generation of
Body-fitted Curvilinear Coordinate System
for Field Containing any Fumber of Arbitrary Two-Dimensional Bodies,' Journal of
Computtional Physics, 15, 299 (1974 .
2. Thompson, J. F., Thames, F. C., Inetln, C.
W., "TOMCAT - A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systeme on Fields Containing any
Number of Arbitrary Two-Di ensional bodies'.'
Journal of Computational Physics, 24,2/4
T1977).
3.
0.
foils.
The "reesace of a zero Jacobian in the field
is not a universal feature in the boundaryfitted coordinate syste, but is peculiar to
the type of configuration adopted for the
trasformed plans in the submerged hydrofoil
solution. TtVUsconfiguration did have certain
212
II
5.
'I
j~.1
General
6. Hirt, C. W., and Harlow, F. H., "A
Corrective Prccedure for the Numerical
Problems,"
Value
Solution of Initial
Journal of Co.mutatiOnal Physir., 2, 114
(1967).
PLA
YSICAL
of
7. Shanks, S. P., "Numerical Simulation
Viscous Flow about Submerged Arbitrary
Hydrofoils using Non-orthogonal, Curvilinear Coordinates," Ph.D. Dissertation,i
State University, Mississip
Mississippi
State, Mississippi (1977).
8.
9.
,Is
...
'1
-"
rsaNsFroflO PLANk
Figure I. Doubly-Connected Region with FreeSurface
C ,r
for
10. Salvesen, N. "Second-Order Wave Theory
Bodies," ProSubmerged Two-Dimensional
Hydroceedings of Sixth Symposium n Naval
dynamics, Washington, D.C. (1966).
11.
c2
PHSKAL PAE
Thompson, J. I. and Shanks, S. P.. "Numerical Solution of the Naviet-Stokes Equations for Arbitrary 2D Submerged Hydrofoils in a Channel of Finite Depth," 094SfEIRS-ASE-77-6, Mississippi State University, Mississippi State, Mississippi
(1977).
"
L
PLANK
THAN$FOMKo
Figure 2. Modified Doubly-Connected Region with
pr,-Surface
Theo.
213
I13
1.26
Figure
4. KarmanTreffz Airfoil
I'
Continued
Figure 6.
. -. . .
,.
,. .; . .' ,.....
(a) F
049-
0.1
(bF
18
191.1
___L
.
) r - .0
1.}1o .o
-/:
Q'~
/surface.
-*jA\
.
-
7.0
- ----
(,),.,.o"-"n.I
Figure 5. Coors.nate system for three Froude
numbers, Re - 20. t - 8.0 - KarmanTrefftz Airfoil located 1 chord below
fre
-'__
.5
,,
-4.0.__
surface.
6.0
0.01
; -7.0
50-
--
-6.0
, -5.0
.455,5.5
Figure 8.
No
7. 0
a-7.01
j4
"I
Wat
(b) t-.
'-
2.0
'
-\Figure
VRe
J4
l~
N.
-
\-'~Trefftz
dinates).
215
-First
17 n-lines
Swim.
22a
Sn,
5,.,S~fl,,nI,.
3.55. SM,,
$I
Figure 19.
215
Coordinate syste at two times, Re. 100, 7 a 2.0 .. Airfoil lorated 0.841
chords below free wirface and 2.2
chords from bottom.
2.0
FRe
chor0sbelc
cls-pofb-.Re-10
084
A-fl
fre-urac
and2
4.0
4.0-
.1c3
Figure 21.
0.16
*..
WI
Figure 24.
I
0.0
-chords
0 .0
0.8
Figure 22.
Tine history of lift, drag and leading edge moment, Re - 100, F - 2.0 Airfoil locatad 0.841 chords below
211
4.0
4.
..
Figure 26.
.-.
1.6--..
.(b)
Transformed Plane
i
o00
-- '_______-4--
0.32,Figure 30.
LIFT CoippICIiwT
0.032672
DRMCCOYRFICIENT
0.743114
'CS4ERTCOEFFICIDIT 0.251998
0.0 .r
Figure 27. Time history of lift, drag and leading edge moments, Re - 100, F - 2.0Airfoil located 1.56 chords below
free surface and 0.27 chords from
battom.
t--
r-
Velocity Verter
ForsSouton
Figuret31.
of free-surface movement..
Figure 2B.Tie
28 1ir6~ history
hord: beow fro-urfca mn
Re - 100, 1 2.0 - Airfoil located
0.27 chords fro
bottom,
foil. T - 0.84
21
Tanlaig
ydo
Figure 32.
T -. 0
Figurot 33.
Lift Coefficient
Hydrofoil
1 -I,
-Plunging
AI
-M.1
Ii
Figure 34.
-.
43t.f
Figurs 35.
Plunging
Velocity Vectors
Hydrofoil (0.001 Amplitude.
0.628 Period)
219j
Contact
tody
Detail of Surface
Region - Plunging Hydrofoil
Period)
0.628
(0.001 Amplitude.
Figure 35.
Continued
0._
-0.1
Figure 36.
0.'
----
Figure 37.
220
Abstract
I.introduction
In a previous paper Haussling and
Van Eseltine (1]discussed the application of
finite-difference methods to two-dimensional
potential flows generated by pressure distributions moving over a free water surface,
Such unsteady problems were solved with a
scheme which combined a numerical solution of
the Laplace equation with numerical approximetions to the time-dependent free-surface
boundary conditions. Both linear and nonlinear
boundary conditions were considered. In the
linear case the Laplace equation was solved in
a rectangular region. in the nonlinear case,
the physical region, bounded above by the wavy
free surface, was transformed into a rectangle
to facilitate the numerical solution of the
Laplace equation,
I. Mathematical Formulation
The Initial/Boundary-Value Flow Problem
Consider a circular cylinder in motion in
water of depth d with submergence h below a
free surface as shown in Figure 1. An (x,y)coordinate system is chosen with the origin
in the undisturbed free surface. The coordinate
system may be fixed or It may move with the
body. It is assumed that the flow Is irrotational and that the fluid is incompressible and
lacks surface tension. It is also assumed that
the surface elevation can be described at any
time t by specifying y as a single-valued
function of x: y - Y(xt). This assumption is
made for convenience and is not valid if the
221
d(x)
Figure 1.
waves approach breaking conditions,
nondimensionalized
The variables are
according to the scheme
(x',y') - L(x,y),
t'- Lt/U.
(1)
p ,pU p, Y'
*' * LU
p I -+t+6ux-(#x2+#y2)/2
LY
R - reststance/pLU
L
-fpn
(10)
ds
ids
ip
lift/PLU
(11)
(2)
(3)
at y-Y
at y - -d
at
at the body surface
Ox
at x t-
(4)
yat
Yt " 6UYx +y
Ot
U/(gL)
/2
(12)
(5)
(6)
(7)
The Transformation
To simplify the numerical solution of the
problem, the time-dependent physical region
(Figure 1), cut off suitably far upstream and
downstream, is transformed to a tim-dependent
computational region which, as shownIn
Figure 2, Is composed solely of rectangles. The
body is apped onto the slit LE, the free
surface onto AB. the bottom of the water onto
the upstream boundary onto ANar4 J1. and
III,
the downstream boundary onto BCand 61. The
and CKEFGrepresent cuts within
boundaries JKLMN
the fluid.
y0
SU#x - Y/Fr I
(8)
222
IIII1II
11
1 11
- l
11
L!11
1 I II...
1- IIIII'
'''"
";''''''""
tl(X.yt)
(13)
=P(&,n,t)
rlx+
fyy
(14)
Q~~~)(15)
J2 (Pyt+QYn)*0
0
I1111 C
where
(17)
(Ytxconstant 'E
2
+ yn2
xC + yj
t2
a .x
1
(9
a.JP(20)
where
XT
4 +#*~~#C
E
ll ly
nt
E#yCn]E/
at nil it
(xE-x#
l
(t 'tn-constant"
t4
(eyn#,E
r
Y
- YC4
(21
t( Ex'
-$nxE)IJwuu(y n# -yC#)/J-Y/Frl
nC
y
-t(Y* Cyt
2
i+(
E
(
x*)
E n-#
at
]/(2j 2 )
(22)
n-n
223
(23)
[7])
can bewritten intheform (ref.
(Jlg2
/2
(16)
and (17)are replaced by central difference
2}=/)
1+,I
(24)
2(i~~~i2
By
equation (24)
can be rewritten in the form
V~l(
*
11(~/
YOs
(25)
~/22~
1Y'~j
1 ~2
+VY
1 l+ilq~
2f
Thus (5 becomesjQ
8
at '1"
/2 i'Yi'j?1
*xi
(2)J'''
v~)/J t n-nb
(27)
1 1
1
10.
1,Y/Jl/2
y1J
5--x-'~
yIIJIY-JIy.~+
2
/ (mij , 'tjj)
-Yi+l'j-l
O8'j
i. anid
'f
j are central
difrneapomtos .(1)
dfeec prxmtost 1)
where
I
with a sector processlnq computer such as the
yn+l and
i,'ne
are computed according to the snoothing
formIla
fi . -fi+- fi.24(ft lfil)O ]1
1
i
nl
(31)
(31)
[(ai,j+xij/2)#i+l
=i~j
+(3j'Tij/2)i-.l~j+(Yi~j+oi,j!2)4i,j+l
(28)
rri~~jI.$)+lil.rldljl)(/2(l
,Pi~)
IV. Results
(29)
Translating Cylinder
and
. n
ilT1
t
n+l
+ 6t(G
+I
+ G0)/2
(30)
1,n
(-t
u
S t 51
(32)
S t
The grid used is shown in Figure 3. The Froude
number Is Fr = 0.566. This particular case was
considered first for accuracy comparison since
Giesing and Smit [12)have presented solutions
to the steady version of this problem.
The evolution of the free surface is shown
in Figure 4. Initially thesurface ispushed
upwaid ahead of and pulled downward behind the
cylinder in an antisymmetric manner. Then a
wave train gradually develops downstream. By
t - 9.6 the surface elevation near the body is
close to the steady proftle predicted by Giesing
and Smith [11], This linear resulipredicts
that the free surface is tangent to the body
surface. Such a solution must be far from
reality since the exact boundary conoitions
applied to such a surface configuration would
predict that no water flows over the top of the
body.
225
t - 2.4
7 .2
V7-
----- L
""G
-2
-1
t.-
- !-
__
x
Figure 4. Linear Free Surface Evolution fo~r
the 7ranslating Cylinder Comipared
with the
Steady-State Results of Giestig and Smith 12]; h 1, d -2.5, Fr - 0.566.
_i
0
-M
ICTEADY STATE, I
I
I
A20
2.4
4.6
13
& MI)-
7.2
t
Figure 5, Time History of Linear Resistance and Lift Corresponding to Figure 4 Comaered
with the Steady-State Results of (ilesing and Smith.
no
3.6
0.011
Y 0.0
UNFILTERED
Z I
4.
-1.5
1.b
0.75
0oe
A 1- 0.3
0 t,._
[.5]
0
.
0--o
.
A
o6
0
#
a.m
6
I AA 0
IA
-.
-eu
-e.u
,A
0000 ' 0000
A
o
[
ooOB
f
Q0
'
-a
I2.
.5
t -4.1
-0.5
-2
-1
Figure 8. Nonlinear Free-Surface Evolution for the Translating Cylinder Compared with SteadyState Linear Results of Zarda and Mvrcus [13]: h = 1.25, d = 2.75.
can exist Incertain situations where an attempt
to reach such a steady state with an unsteady
numerical calculation for an accelerated body
would fail because of wave breaking during the
transient period. Such steady, nonlinear
solutions might have to be approached in a
different manner. It seems quite likely that
at least some of the nonlinear problems considered here fallinto this category.
1
-
nPi--
ILOWER
can
be followed to t : 4.8 before features
develop
which cannot be adequately
handled by
the numerical scheme. Figure 8 shows the freesurface evolution aleng with the steady linear
surface profile comput,, by Zarda and
Marcus [13]. A trough develops downstream
from the body. As the gap between the body and
the surface narrows, t.', flow speed in the gap
increases. This increased flow speed leads to
increased downstream convection which prevents
the trough from moving to a position over the
oody as it does In the linear case. The
computed pressure distribution on the body at
t - 4.8 s compared with the linear steady-
SURFACE
U R
SURFACE
-3.5
0.5
3.3
NTV=
FREE
SUR AC
41
-.
-4
-- LINEAR (ZARDA
.t L
MARCUS)
..
-1
Figure 11. Nonlinear Free Surface Evolution for the Translating Cylinder Compared With
Steady-State Linear Results of Zarda and Marcus: h - 1.S. d - 3.0.
no9
Swaying Cylinder
In addition to the translating cylinder, a
saying cylinder is considered, A fixed
reference frame is used so that 5 = 3 in (21)
and (22). In the nonlinear case the finitedifference grid must deform in response to both
the surface waves and the movement of the body.
The center of the cylinder is one cylinder
diameter below the undisturbed surface, and the
water depth is 2.5 diameters. The characteristic speed U is the maximum
speed of the
cylinder. The dimensionless frequency
(frequency *L/U) is 4 and the Froude ntAber is
Fr - 0.354. The horizontal position of the
center of the body is
x - 0.25 cos (4t)
(33)
.. ._i._ [I I~i
Im i
VI. Acknowledgment
This work was supported by the Numerical
Naval Hydrodynamics Program at the David W.
Taylor Naval Ship Research and Development
Center. This program is jointly sponsored by
the DTNSRDC and Office of Naval Research.
References
1. Haussling, N.J. and R.T. Van Eseltine,
"Finite-Ditterence Methods for Transient
Potential 'n-, ,ith Free Surfaces," Proc.
of the First International Conference on
Numerical Ship Hydrodynamics, David W.
Taylor Naval Ship kR.earch and Develooment
Center, Bethesda, Md., 1976, p. 295.
2. Nichols, B.D. and C.W. Hirt, "Calculating
Three-Dimensional Free Surface Flows in the
Vicinity of Submerged and Exposed
Structures," J. Comp. Phys., Vol.12, 1973,
p. 234.
3. Ohring, S. and J. Telste, "Numerical
Solutions of Transient Three-Dimensional
Ship Wave Problems," Second International
Conference on Numerical Ship Hydrodynamics,
Berkeley, Calif., Sept. 1977.
4. Thompson, J.F., et al, "Solutions of the
Navler-Stokes Equations in Various Flow
Regimes on Fields Coataining Any Number of
Arbitrary Bodies Using Boundary-Fitted
Coordinate Systems," Lecture Notes in
Physics, Vol.59, Springer-Verlag, Berlin/
Heidelberg/New York, 1976, p. 421.
S. Shapiro, R., "Linear Filtering," Mathematics
of Camp.,{tion, Vol.29, 1975, p. 1094.
6. Longuet-Higgins, M.S. and E.O. Cokelet, "The
Deformation of Steep Surface Waves. I. A
Numerical Method of Computation," Proc.
Roy. Soc. Lend. A, Vol.350, 1976, p. 1.
7. Thompson, J.F., F.C. Thames, end C.W. Mastin,
"Automatic Numericl Generation of BodyFitte,,
Curvilinear Coordinate System for
Field Containing Any Number of Arbitrary
Two-Dimensional Bodies," J. Comp. Phys.,
Vol.15, 1974, p. '99.
INEAR
-O -
I f-I
--
1,
tt- 1.8
2A
tJJ2.L
t
--
-0-
v-
t- &0t
.
"
-4t
I
2o-
0*
Fre
ura
I
Evlto/o
e
-.
wyn
yidr
Fiur Fre
1. url
T
Figure,
1
I-
uin
or
heS
lidr
3'
.,.
'
II
I3
1~
~ fl~
t 1.8111
Figure 13.
232
8.
233
.~
DISCUSSIONS
of two popc
Invited Discussion
independent variables of the coordinate transformation are then interchanged so that the flowfield computations can be conveniently performed
on the fixed (&.n) grid which does not change
with time. The flow equations to be solved must
also be transformed to this computational grid.
Although this step introduces cross derivatives
and complicates the solution of these equations
somewhat, the advantages gained by performing
the calculations in the transformed region
include the convenience of a uniform rectangular
grid and greater accuracy in applying the boundary conditions along straight lines. There is
considerable flexibility in the choice of the
configuration for the traisformed region which
depends both upon the geome,-v of the physical
flow domain and the extent to which grid-point
density is desired in critical locations. For
submerged-body problems, Haussling and Coleman
chose an H-shaped transformed region which
exhibits certain advantages over the T-shaped
region employed by Shanks and Thompson.
Joanna W. Schot
David W. Taylor Naval Ship Research and
Development Center
The two papers just presented demonstrate
quite impressively how far finite-difference
methods have evolved in dealing with initialbounoary-value problems for two-dimensional
irregularly shaped flow domains that deform with
time. The authors circumvent the well-known
difficulties associated with the use of rectangular coordinates for computing free-surface
flows around arbitrarily shaped bodies by employing numerically generated curvilinear-coordinate
systems in which a coordinate line coincides
with each of theboundaries of the physical region. This means that the curvilinear finitedifference grid must be numerically determined
at each time step along with the solution of
the fluid-flow equation
It is especially .,eresting to note that
the technique for generating such curvilinear
coordinate systems is very general (conformal
mapping is a special case), and is independent
of the flow equations to be solved. Thus, it is
.appropriate that at this Conference we are exposed to the results obtained with the use of
this approach for two different types of flow
formulations. Shanks and Thompson have solved
the difficult Navier-Stokes equations of viscous
flow past hydrofoils moving in or near a free
surface at relatively low Reynolds numbers. In
a complementary effort, Haussling and Coleman
have treated the infinite-Reynolds-number case
of unsteady potential flow past submerged circular cylinders with both linear and non-linear
free-surface conditions,
While there are various ways of setting up
the numerical procedure for defining the curvi near grid system and performing the flow calculations, both papers here use an elliptic system
of equations with Dirichlet or Neumann boundary
conditions to transform the physical flow domain
in (x,y)-spatial coordinates into a region composed of rectangles in (c,n)-coordinates with
constant mesh size. The use of elliptic aquations for the coordinate-generating scheme
appears to be a natural choice suggested by
the extrmum principle for certain elliptic
boundary-value problems. The dependent and
234
In both papers, the systems of finitedifference equations approximating the coordinate transformation equations and the flow
equations and the flow c tuations are solved
simultaneously using the asy-to-prog-am SOR
(Successive Over-Relaxation) method. Much faster
methods could be applied, and are indeed under
investigation by several researchers, for example
, GhIa and U. Ghia of the University of Cincinati. Haussling and Coleman rr'er to their work
on alternating-directior implc(it methods in
their paper, and they have recently reported
2rjr
met'~ods 121.
ti
K M
1,
Mra
r
"
"
N
.
AKI.G
_._
'a
30,40
to
INFT/Ut
WEDO
Discussion
byNl-s-Salvesen
of paper by H.J. Haussling and R.M. Coleman
The steady-state nonlinear problem of uniform flow past a body may be solved by two
approaches: (1) It may be solved in the time
domain as an initial-value problem advancing in
time until a steady-state condition has been
reached, and (2)it may be solved as a steadystate problem where the free-surface shape is
initially assumed and then through an iteration
scheme systematically changed until the freesurface conditions are satisfied. Since it
would be extremely difficult to construct an
iteration scheme that would work for the thr,:edimensional ship-wave problm, it might seem
that the initial-value approach would be the
most suitable for three-dimensional problems,
even though we have had more success with iteration techniques than with the initial-value
approach for two-dimensional problems.
However, as shown by Haussling and Coleman,
the numerical time-domain solution may often
break down before a steady-state condition has
been reached.
rest without wave breaking occurring at intermediate stages and therefore it seems likely that
any numerical scheme modeling such cases would
also have to break down at some intermediate time
step. I would like to use this opportunity to
suggest some initial-value time-domain approaches
where this wave-breaking problem can be avoided.
For the two-dimensional case of a submerged body
as Investigated by Haussling and Coleman, one
could, for examsple,
let the starting condition
be the uniform flow past a daeply-submerged body
and then slowly decrease the submergence until
thedesired condition is reached. One could
also let the starting condition be the uniform
flow past an infinitesimal thin body, and then
slowly increase thethickness of the body. Such
an approach would be applicable to both two-and
three-dimensional bodies. However, these approaches have the disadvantage that the geometry
of the problem would change with each time step.
Therefore, the best approach seems to be one
suggested to me by K.J. Bai,namely, to consider
the body as a porous medium. One would start
with uniform flow, going completely tnrough the
body and then in the time domain change the bodyboundary condition so that less and less fluid
goes through the body until it finally becomes a
solid body.
Author's Reply
WysJWIIfisling and R.M. Coleman
to discussion by Nills
Salvesen
We would like to th.nk Dr. Salvesen for presenting some interesting experimental resulcs
which support the plausibility of our numerical
solutions. To continue his discussion of methods
for avoiding transient breaking waves we point
out once more that the boundary-fitted coordinate
systems can handle time-dependent geometries.
Thus changes in body thickness or depth of submergence could be treated. However, the most
satisfying solution to tne wave-breaking problem might evolve from the current studies of the
physics of this phenomenon. With the development
of mathematical-nuzerical models of breaking
waves calculations could be carried out beyond
the time at which breaking first occurs.
236
Abstract
Available and developing techniques for the
numerical solution of three-dimensional, fullycavitating flows about hydrofoils and other
bodies are reviewed. Three areas are examined, viz. , linearized methods, methods based
on matched asymptotic Expansions, and fully
nonlinear methods. For the first two areas
there are presently available two specific techniques usable for quantitatively accurate design; interestingly both techniques rely on the
use of two-dimensional characteristics o" the
flow applied stripwise in the three-dimensional
flow field. The area of fully nonlinear methods
is still developing. The finite element method
and the inverse, stream function and potential
method have both been applied successfully to
three-dimensional free surfa:e flows,
Introduction
It was just over eleven years ago that
Widnall (1966) published her numerical simulation of three-dimensional, fully-cavitating
flow about hydrofoils. That work and work preceding it incorporated a number of limiting
Linearized Methods
Widnall (1966) derived a linearized threedimensional, lifting-surface theory for fullycavitating hydrofoils of finite span in steady or
oscillatory motion through an infinite fluid, The
planform of the cavity was assumed to be a rectangle of finite length and of span equal to that of
the foil. Widnall (1966) argued that the predicticn of lift and moment on the toil is not aensitive to the assumed cavity length when it exceeds
about twice the chord of the foil. It s well
twic
th
*This work was supported In part by the Naval Sea Systemrs Command General Hydromechanics
Research Program, Subproject SR 023 01 01, administered by the D~vid W. Taylor Naval Ship
Research and Development Center, Contract N00014-75-C-0277.
237
IL
W ~ea~~
236
Solution of the far field problem is straightforward. The key idea of Shen and Ogilvie (1972)
was to solve the two-dimensional near iield
problem exactly, I. e. , without introducing tinearizing approximations. The method of matched
asymptotic expansions was then used to maich
the near field and far field solutions. The result
is a uniformly valid nonlinear solution matched
(in this case) to the second order in the small
1
parameter f = AR- . In addition, this method
yields an unambiguous definition of the height of
the planing surface above the undisturbed free
surface at infinity.
The hydrofoil problem corresponding to Shen
and Ogilvie's planing problem was solved by
Furuya (1975b). His objective was "to provide a
simple yet accurate method for design of supercavitating hydrofoils of large aspect ratio near
a free surface, having practically no limitations
on the admissible foil profile and angle of
attack. " In the context of matched asymptotic
expansions as described above, Furuya (975b)
employs a two dimensional nonlinear freestreamline theory for the near field flow region
and Prandtl's lifting-line theory for the far field.
The small parameter c = I /(Aspect Ratio) is
used as in Shen and Ogilvie (1972), with (Aspect
Ratio) = (span)Z/(projected area of the foil on the
horizontal plane). The matched asymptotic expansions are taken in the limit as
..0.
The
effect of this approach, it should be remembered,
is to eliminate consideration of the '-'havior of
the flow near the foil tips. Thus, the cavity
width (spanwise) is equal to the foil span exactly
at and near the foil. Furthermore, for the far
field problem to be a lifting line flow, it is
necessary that the foil plus sit3
chord be
small compared to the span. Accordingly, the
theory is valid in principle only for "short"
cavities.
239
(5)ikl~iina
P0
/a
Se.e
Fiur
t~ds~stwDmnsoalFo
Schemati
E
17b
trke
uua
te
j oA-f
Coodinte
Two
3ofSchreeiDimensionalield,
Figure
1. Schematice
() Fl
Ac.
wofigrtin
k-
Diesoa
Flwi
FuruCorinta
Te
tece
corNearpandin
FarFiedaBounda
fbor
d '
Hydrofoil
Flow~~~rblm
)4.
roVluem
TPr~) Aerfter
(1975b)b.,
Furuuruy
Two-dim.eoalanonlar
Largeolem
tal
LAAfter10
er-fiedr soluo
Asec
parAfeter
Hydrofoil
Furuya (1975b)],
The tue
rgeittin
dumeptrhs is
rltd
The
~ ~ ~ ~ ~ nerfedcniuaiousfudb
~ e- ~ rsaof
a
~ting ~~ ~ ~ie. ~ m.ifiel
(-0wttecodhl
~ ~ ~ a
accmplshe
bydesribngthenea fild n
Figurve
(xlf sy/,
i af).
th cas ofShe
Q. ) tcheaine ar Fd lw Flon
an
o utnear
oh Fuuya (97~a)whichemplos
theTThe
doubleaprl voex odeluato
(i.
fon3).
tin
hechod
0 wih
The corsonigbonay-au*polm
-~F
Su
o-f( NerFel.
i
ol
fxe
T isto
aieedb
c m
in. heurnnears
and asifiel
f
solutionfIr term a tansbtrctintei
dfndsubmergence depth ()s
nreated to he
rsth problem
oy inatlodimen ion).
teloato
ofth
sagnti
tr
amine
rua219
5b
MAWRO
CJVITATION
e
"
cavity sheet et downstream infinity in the twodimensional solution to be equal to the "down.iash" angle obtaine in the three-dimensional
solution.
..
..
t
ngprocedure.
r-tLFi
Furuya (1975b) describes a numerical iterative procedure to solve the system of five
nonlinear equations described above. His procedure converges rapidly and stably in 4 o 8 - 3
iterations to relative errors of less than 3x0
FI
i:
.
.0
ri
-''
o. a s
'=
...
01o
.OS 0a.
SFigure
-
..
____
'treated,
".
""-
CA
*
Figure 4.
K 0cavitating
0
a
24
L,
An Overview
There are several major categories of numerical techniques applied to free surface
flows, including finte-differ-nce and finiteelement methods. Amongst the finitedifference methods we also find formulations
in which the problemi is solved in a stream
function and velocity potential space or in
which the formulation is based on the distribution of singularities on the free surface of
the flow (normally called a boundary integral
equation technique).
sional and axisymmetric ideal fluid flows involving a free surface. The trial-free-boundary
technique used involved assumption of the location of the free surface, solution of the resulting
well posed problem, and relocation of the free
surface according to the free streamline bounda ry condition.
In the next two sections we follow up on two
methods. First. we review a three-dimensional
formulation of the inverse stream function and
velocity potential by Jeppson (1972). Second,
we deucribe our own progress in the application
of the finite element method to three-dimensional flows.
The Inverse Method
Working independently Jeppson and Brennen
both developed inverse solution methods for free
streamline flows and presented these results In
their Ph. D. dissertations in 1966 (see Brennen.
1969, and .eppson, 1969 and 1970. for example).
Jeppson (197Z) and Davis and Jeppson (1973) subsequently developed an inverse formulation for
three-dimensional flows in which a velocity
potential and two stream functions were used.
By changing the conventional roles played by the
variables of the problem, the inverse method
converts a free surface with an unknown position in physical space into a plane of known
position in the inverse stream function and
velocity potential space. The major disadvtntag# of the method is that the shape of curved
solid bodies cannot be exactly prescribed in advance in physical space. However. ,eppeon's
method has been successfully applied to a
three-dimensional groundwater flow with a free
surface and to fully-wetted flow past a body
moving beneath a frue surface. Accordingly. it
seemed appropriate to review here the fundamentals of his formulation.
of the c
irtesean
coordinates
d.
a) be the
-L8--"
0-z .dx ' - b
potn-_
velocity
2 2X
and define( a y,
dependent variables
tial 0 and two additional functions 6P. , #) asa
the Independent variables. The two streamThsaraseofculdnnier.ist
funn.tions defined by Yih (19S7) us surfacesThs
rasto
opldnoierist
normalto equiptentil surfaces and angeorder prtial differential equaons. When ombined with appropriate boundary conditions this
tial to the velocity vector (such that their inset comprises a well-posed boundary value
tersections define the stream lines of the flow)
problem. This coupled set of first-order equameet the necessary requirements. As a result
tions must be solved simultaneously and has
aean a major stumbling block to more aggres_
!!f
ID.ye
By as
"a
S-
as
B
x
aBrennen,
8-y
by
Bx
080
Sh
By
ex $I
__
(3)
(4)
(4
Ox
By
by
Ba
Se
3s:
tial equations."
84A
0.0
&a
grad 4 x grad4o*
(5)
..
fied.
The surfaces formed by holding %P and
*
constant are orthogonal to equipotential sur-
3f..e..
orthogon
.3-space
for (4, *,
-----------
'
)..
vera.,
uctions
(,4,.4*),.
Vial.,,L
5040.0),
Bx . !
It ." AL
85
SO
*. ,.,*)d
|6)
Figure
84* 34-
!a
a.
0
OZ s
34
(7)
(After
Spaces.
3(1973).
243_
__
I -
finite-difference grid.
S..prolate
u,
'......
Ccavity
/
a,
F ,re 8.
.-
244
Ce
me*
m~
r 0
0= N 1 ,N-N
dx
dz
dx
-80/az
2i0
=N]
I
to
I
(15)
(9)
u = -P "lax; v = -80/8y:
(10)
space.
(I
2
u
2
+v
+w
= q
= constant
(1z)
(13)
qcs
te4.m Co-ow.dhs
Ca-adkw
Laceg(WNurol)
Figure 9.
Now the functional in Eqn. (14) is minimized with respect to the nodal values oi
within each element (Huebner, 1975).
Thus,
|
8-'=
,I
"
method formulation-
9N.
Functional:
I N, ON
MI
|
dV =
(16)
f
fl
v z-
NJ
'-
-!T 2
By
'
2or
h] jol s - 0
azON
(17)
(I)
Within element:
2f4
of
the 0 t or 4P derivatives along the flow
boundaries.
v.
20
20
. So,
S .
20
Ni(C, 1. ;)a.
S(R, I,)
z0
Ni(4,.,t.)0i
4(0,,,)
10.
(,9)yFigure
(19)
The boundary conditions for a fully-cavitating flow in a water tunnel are straight forward.
With a Riabouchinsky model, the downstream
boundary is an equipotential line. If the potential downstream is taken to oe aero, then when
a free surface shape is known (assumed or calculated) the potential can be computed on the
4s in the 3acobian
13]
aries, except upstream from the plate. Typical boundary conditions are illustrated in Fig.
WOEa
ByI84
8xi81)
By/il)
:/9/
By/8
8I8
xet(
10.
/at
(20)
If F. W,
\2
Poo" PC
I
dxdyds
"I
dtd
l3ld
(21)
and
AN
FNI
ex
--
ON,
,
J]
ONj
-(22)
(N)
by
With
L 0
ON
J
(Z3)
/u
large.
The
d,,
dv dt
1',
LI
,
-_
SYJ T
b..
P1W
L4
L/P 2.
1a).2
-
.qI
ployed in planes x = constant to locate the element transverse mid-nodes which are missed
by the in tegration process.
f. Establish new values of 0 on the free
040,
-Some
T~, Fim
T
surface.
Only
0.5
o.e
02
ro
.WAhns
.lellsp5
-
0
4
10
12
w/P
Figure
Street and Ko (1977) applied the above technique to two geometries. First, they reproduced the geometry used by Brennan (1969). via.,
a Riabouchinsky model of flow past a circular
disk in a circular tunnel. The finite element
results for 9 and cavity radius B agreed to
within 2 percent with Brennan's results when
the physical geometry was specified. Second.
they examined flow past a disk in a square water
tunnel, demonstrating for the first time (a) a
fully nonlinear. three-dimensional cavity flow
and (b) the effect of three-dimensionality on the
wall effect in thic flow (as compared to that in
the axisymmetric case).
Prognosis
The prognosis for numerical simulation of
thr e-dimensional, fully-caviiating, steadystate flows is good. Jiang and Leehey (1977)
have developed a linearised method which is in
good agreement with experimental data and is
not limited to large aspect ratio or short cavities. The method of Furuya (1975b), while
having such limitations in principle, works well
ovtside those limits and is nonlinear with respect to angle of attack. Neither of these methods accurately models the tip regions of the
flow, but this does nct seem to be a malor limitation at the moment.
nonlinear to simple pure drag flows. AS iiustrated herein their simulation can be extended
and is applicable in principle to hydrofoils in
lifting flow. This method directly handles the
tip regions, but may experience difficulties at
low angles of attack due to the rapid changes in
flow near separatior. and the extreme curvature
of the free surface there.
Th(. inverse method of Jeppson (1972) and
Davis and Jeppson (1973) deserves further
associated
study. In spite of the difficulties
with solving the governing equations, the method
has been successfully applied to free-surface,
non-cavitasing flows. An extension to cavitaving flow based on a Riabouchinsky model appears
feasible.
A method likely to make a major contribution to fully nonlinear flow simulation is the
boundary integral equation technique. White
and Kline (1975) used itsuccessf.ally for axisymmetric free-surface flows. Larock (1977)
presents an application of the method in these
proceedings. Finally, Hess (1972) has demonstrated the power of the technique in fullywetted, three-dimensional, lifting potentidl
flows,
Acknowledgments
The finite element work for three-dimensional cavitating flows reported herein is a
joint effort of the author with Peter Y. Ko
under the sponsorship of the Naval Sea Systems
Command General Hydromechanics Research
Program, Subproject SR 023 01 01, administered
by the David W, Taylor Naval Ship Research and
Development Center, Contract N00014-75-C0Z77.
References
1. A. J. Acosta (1973). "Hydrofoils and Hydrofoil Craft" Ann. Rev. Fluid Mach.,
5, pp. 161-184.
2.
A. H. Armstrong and J. H. Dunham (1953).
"Axisymmetric Cavity Flow" Arm. Research Estab., Kent, England, Rap. 12/
53.
3. C. Brennen (1969). "A numerical solution
of aulsymnmetric cavity flows" J. Fluid
Mech., 37. pp. 671-688.
S, T. K. Chaa end B. E. Larock (1973a).
4.
"Fluid Flows from Axlsymametric Orn(ices and Valves" 3. Hydr. Div., Proc.
ASCE, 99, pp. 81-97.
N. S.T.K. Chan, B. E. Larock andL.R.
Herrmann (1973b). "Free-Surface Ideal
Fluid Flows by Finite Elements" J.
r
., Proc. ASC. 29, pp. 959974.
A. I.. Davis and R. W. Jeppeon (1973).
6.
"Solving Three-Dimensional Potential
Flow Problems by Means of an Inverse
7.
0.
8.
0.
9.
0.
10.
D.
240
22.
23.,
24.
Z5.
Z6.
27.
28.
29.
30.
31.
32.
33.
Y.
T. Shen and T. F. Ogilvie (1972).
pp.
44
Abstract
Introduction
Considerable theoretical and experimental work has been done on threedimensional supercavitating hydrofoils,
but the agreement between the theory dad
experiment is not fully satisfactory.
It
and an elliptic source, vortex distribution along the span to transform the
three-dimensional problem to twodimensional equations for rectangular
Linearised Theory
aspect ratio.
large
A matched asymptotic
The
250
is
shown in Figure 1.
source strength be zero beyond the cawity trailing edge and the sum of sources
CAVITY4)
dimensional
CONTROLPOINT
AT((k i
/wI
0-
steady
ten as
(X
o-- 3)
, )~~SriE
fJJ
q(x,z)
qI
point (x,o,z).
-___
()
'p.
2"
u,-
result
'-(vo,))-vx,-o.)),
( s,),,L (
(+o,)
(4,), )
us
(4-)
r(J,-.,4).
U.
,.-O.)
P(,to',),
PC-
9I,)),
(Z)
surface location.
where h(x,z) is
"
e(,S)4Ed$,
wetted surface. Sc
251
wJ se integration becomes
, 1(t)
'I$)
(,
0.
(7)
JSJ)I
A/0)
1) Ais,
55
(vU%.s-
a -as
1w)
edge,
one-half the chord
ing of have
the only
rest of tho elements on spacthe
foil.
2a(24
CI
,(8
I t.
'
+c~
~
4
C a e'
252
()
(3)
for z>z
IT
OX
2a'
+ 2--.f-1,
(9Sdncavity
is the source potential. The
P. due to the line source ((x ,O,Z ) to
(x2 ,o,z2 )) can be calculated by the line
t,
I,,
where
integratio.,
(I.A)
i Oz)
ZW
W1
be
((-+(
)P
()
Table 1
No. of element
along the chord
b
(24+6
12
oa the foil
(i) l
No. of element
7-
in the cavity
behind the foil
a to 14
No. of element
along the
(14)
~t
emispan
253
Gt
o 60"
o e0 , MMe
4..
s.
',
I
0
a *.I,'
bo
4.
z 4
.l
10
.o I1
a
....
__
__
_|
Ia,
10
pl'ate
ZI
M -5
..
I
0~~~
i i.
1,
Ep~c
aa.S
Leehey 5 StenltaQw
1.41
a0
Ew.15
a
POnM.1
12
15:::
aItWq a
U..
o..3
to
-a
-10
2.
&
W/o
3.0
2.0
1.0
0.
oT/a
Figure G. CM/a
Figure 4. CL/a vs. co,
vs.
a/a, Ai-3
Mt-3
/.
experiment, the
In Maixner's
cavity pressure was measured with a foil
surface pressure tap. A noticeable
in
the
and (Figures
moment
"hook at " higher
foundangles
of lftattack
data
3 & 5). Ram effects on the cavity
pressure measurement, due to the dynamic
pressure, were further investigated on
"
0[ ________
0
1
1
3 4
Figure 7.
--
vs.
L/C,
UC
)R-
headtub,
an
onthefoils
Ik
,el
im,.I
Figure 5.
a. /a
A-s
to/ta,
I----
Figure 8.
L/C,
t.
..
/.'
is
*1
6 Ilhu*d:
,.ma,.
Figure 9. cc vs. av
downwards into the cavity from the upper
tunnel wall so that it was parallel to
the oel1surface and pointed towards the
leading edge, away from the impinging
re-entrant jet. Figure 9 shows that
the readings from the fol surface
pressure tap are consistently higher
than the measured cavity pressure,
especially at higher angles of attack
and shorter cavity lengths. If the
improved cavity pressure measurements
had been taken in the experiments of
Maixner, the discrepancy from theory
at higher angles of attack and shorter
cavities would probably hlave been smaller.
V Conclusions
The discrete vortex and source
method was developed for supereavitsting
was
hydrofoils. Tho cavity ]-.!,gth
iterated to got a uniform cavitation
number over the cavity planform. The
lift and moment coefficients for supercavitating hydrofoils of elliptic planform in steady flow was performed and
Acopared
more accurate
prediction
lft and
well with
previouso experiments.
momtent coefficients was obtained by the
present numerical method than with
existing asymptotic theories.
Acknowledgement
1!leading
This~~~~~~~ reerhwscridotudr
ThNv hir
sachs
c.aid,outenra
edge, awa
frau the impingingIi
Hydroaschanics
Research
Progra-m, Sub-EnnernN
exn
.1.97A
x;r
na
ta
97AnEpim
10RinrR.M
Investigation of Wall Effects on Supercavitating Hydrofoils of Finite Span,
Reference s
1 Xfremov, 1. 1. & 8oroka, R. A. 1975
tuPrOX at Comtatit wa parallelto
th oal
urfund
Soan in t
hnite
Fluid Dynamics.
11 NishiyU,
T. 1970 Lifting hine
Theory of upercavit tng Hydrofoil of
Span Pogr, Vol. 50.
jNo.
83481-2
267
1. INTRODUCTION
Linear and nonlinear models of supercavitating foils
14
have been investigated extensively.
' All investigations
involve a considerable amount of numerical work. Thus the
advent of high speed computers naturally requires efficient
numerical methods to be applied to the solution of the flow
field and the design of supercavitating loils. Recently a large
effort has been exerted toward numerical computation of
5
16
cavity flowl either
by the singularity method
the finite
17
difference scheme
or by thefinite element tlechnique.rt
This effort seems to be particular;y fruitful for threedimensional flow, highly complicated boundary, and
consideration of nonlinear effects. In the present study, "n
application of numerical technique to a linear twodimensional supercavitating foil is considered, especially
from a design point of view.
Fourier series have long beer.used in linear airfoil
19
theory.
And a theory of hydrofoi-airfoil correspondl
encet1i has made the Fourier series useful for the linear
theory of supercavitating foils, lowever, when the boundary
conditions become complicated as for foils in a cascade
Fourier series become cumbersome'when carrying out
computations. Fortunately, as ir other applications of
Fourier series, such as to informlion theory,
the use of the
t0
Fast Fourier translorm (FFTt technique
greatly assist
numerical computations.
In tlhe present theory, in order to use tire FFT
effectively, many physical quantities are convenieritly
represented by Fourier series and its coefficients. It is
demon.raled thai FFT computation can be performed
accurately in a relatively hurt lme, By a simple change of
the transformation fur,.lion. le meihrd can be earl),
applied it different boundary conditiliss such as those for a
infinite medium. beneatl
a free surface, or ii a c ascade,
lie linear theoryofsupercavitatiig toils is particularly
5 1
pressure distribution,'
0 this does not guarantee satisfactory
foil performance. One other possible design method would
be to specify the favorable camber shape first and then
determine the angle of attack and the leading edge cavity
thickness. For this purpose, it is necessary to have a method
for computing the flow field about a supercavitaring foil of
arbitrary shape. The significant physical quacirties of the
flow field include not only the pressure distribution, the
lift and tile
drag coefficients, but also the shock-free angle
4
of attack. and the foil cavity shape.
The present study deals with these problems, exploiting
many advantages of FFT. A particular advantage is that the
shock-free angle of attack is readily supplied by FF1.
Numerous results o computattvns are shown for "
c of
a cascade. The design of a sup'.aviiirng foil isaccornplislicd by combining three different elementary foils. a
shock-free foil with a given camber 5shape, a flat plate witll
an angle of attack, and a point drag to increase the lcadirrp
edge thickiess. The relative effectiveirs of the angle of
attack and the pointdrag in increasing cavity thickness is
examined numerically.
The present method and tile
pr igi may i'i
flriprIer
utilized effectively forthe desigin f ..
itsting roil
for a high speed hydrofoil or a superc:ril-iting propeller.
2. FORMULATION W[.iROBLEM
A supercavitalifig toil with i irnfinitely lung cavity is
first considered rear a free surface or in a cascide. 'lie
angle of altarik rif tie fril is 'ui rod the flow periurballoin
due hlitl :oil i assumed to he silalL. i,- A ilicar theory
is consideed to be applicabi. l,,
I ii. rn-alii, I riM'
statical boundary for Ihe foil and svi i; nir"aiong the s axis which is palnel to tie velocity at v - -where the velocity is unity. Four pissible flow geometries
tif a foil mt,
s
ire trrir
",l l rlt lrrfr ;In, cat.:l
'-r
vt Figll-",
I
For convenience, the complex perturbation vel .ity.
v
u-iv, is changed to a modified c'ompleil,
vd,!
Fitji I t, 0u2 - IV-_) where t i the , ,.' - ,trir
derinedt by tife pressure u 5
"t. P. arvi the pressufr
it
tile
cavity P,
P
"P
Pv,
l
288m
ql,
mm
,,m
r.
B..
A..
_____D____B__
A--.
Vi,
A--
a-0B.
.0
u-
0 VIA
1
B..Yi,
E_
rX
1)
il
2
iPLN
1
or
where
=K+
8-.X
u-0/2
.(.) A
lig
d..= K .
..
Iy
PLANE
-1(2
and
E.
uO
-t
;q)
(Ct
Figure I
dx
+,
in y tanl I
a2
co, -y/4 I -
4111y+ a
'
- l?
PLANE
;-0
;Ip
-1
-+PLANE
rl
B.
-.----
intle Ca sy Cascade
Figure 3 - An I
C
8U.
A.-
O0
-0o
EC)
o.a
Ct1,0-)
Il0)
a.0
A..
(11 g - w 4 iv PLANE
KC
dz
j -
0 '0
4*v-
0a-K
._''
t)
*0*/
,-,
A4 ..
C 'u,42
A3 u*"
+ a lo
whicr+
K.
_((
and
K -
9) a"
ly PLAN&
Ih[
K-
A.--
5t1t-
e
IC
Figue 2 -
A SupeYi(
03 1-aI2
.1
t-
(Ct
Fir
a Fm Swfb
250
-3
,.n0/2
A,
U..
PlAsqP
iANE
Iunl Foil Ab.c
-1
Illl
'PLANE
4 - FIeate
avityCacde
B.csn()
".0
i
0> t-10l-Cosa)>-,
Acot!+,PAnSin no
2
2'-
(I)
0>t> -
~<
-~e
to
(- 202K di
22
I - 2at Siny +
(2)
Isin Oda
dt
0 <6< I
K(A cot~+
n--A+Z A cosnO
rx
(3)
n1I
dO
_
+a(i - cos 0) siny + a (I -CosO9)
and4
G/2 - v/2 - u
vo.-
v
(3a)
A av_ -A,
Bn cos O sin0) dO
A~sin nO
+f
n-0
n.1
v.
nenl.
0B+nj..
32e1
(3b)
a-0
-45(j)
(4)
-cs(
)0
0 o 0Od
n+IOj2.Bacn d
5 .0
4+-BA,
i
- 2wA /IB+!B
~B,,(An,
2
+~BA
221
I A,
(6)
n-2
The moment coefficient of the transformed airfoil Is
2
0'l2a - 7u)
ise
in a casicanle,-
517+03t,
I -2 sin
04
+
{Ail -cos~t)f .
Asin n
cot rig
Baxis
-00
JEA
I5-cI
B os.
.
Tabir
Flow~
Charaeamas oi
rniet.
TT
ingle toil,
i
toad distribution!
-2U(x)+o
Airfoils in
I
maI
nfintjt
mediu m
i",
the trnsfortmed
,Cscatdes
plane
wlor
-2
hi
in'ii
)4t
Lift Coefficient
Drg C oefi i n
Lea
la in y
upper
l~ds
boun
c"
otpr
-from
of
Xc-
shurrey'
tt17
h =the distance
the leading
to the free
1.1edge
d-
iin -1CLXQ)
I
----
--
or mornt About
1theleadingedge
--
No.
~. n
- CosIn + Il)f}
n1
Ij co I.
n Co
An- 1 0~
f
Transforma-
+a.M
Chordlength I
KI
~cuE
B.nA:.nO a 0,l- I)
n0~~
&i'Yk
ev
+t
idih
b velocity rtormsal
gote
0) canbeobtaimed by
A~~~
(Bf-1 )
Ga
oldt
stagger
angle
v,
(.1,Surface
-r
(.rvit atIA
+ a
+nl7
A,Qa~
I
1
uA
~A,.12+
261
FAn
in RO Ain$
In
<-I
! r
v-v=-
Anjcost-l1l0-costn +tIt0
AII+cOs61t+.I
a +adO
=-
A,
1
a + cos 0
a
17
case.er
nat plate
A=
in
/I
- I
On +
A,
where
n-4K
=A.
=
___
Aa-I-Antl~an
v
7
with a = -
(14A)
4K
2a sin '
n I
(BO
+! u)(B,
tl)
+ 4K
2 > I
A=
In t > 0
l,-
(AO,- F i / 4Ktl
16)
=
a+I
vA--
nv]-1-"
+LAni -
tOI+
6. CAVITY THICKNESS
with al I 4 21>1I
The velocity at x
and FM-
as may
4wK
be expressedn in terms of C1
f-a si y
_o a
2 4
T xnk Cos7
f(2)
2{
4K La
4
I,
sin + 2
n 7 'T
',
--
.
2
4K
where
-
I AA]
2a
n- z
,- 0'
-r
it1 )
5I-(Av.-'t)
,
si
n-
iltackl. A, is given by
where v2
is the y component of velocity at
- dun to
just the flat plate.
Mle fl-plate contribution :o cavity thape is obtained trom
F, -4K An
( 1I1
A
{:a,,.,-
(B4+
B)lei
+A-(Ilo
)}t4K
22
vx) dx
q)
nol intended to be mathematically optimum. This optimizalion aspect is investigated in the following section for a
supercavilaling foil in a cascade by systematic change of
dx
dt
2variational
Si
d
I - 2av
I 1X)
,Y+ .,2t
-.. lan
cosy
sn
+
Cos7esy
Y/CX =
y, + y.
Y+
"
t f, + kvi
=c i
(2I1
I(x iQ)
20)
with respect to x.
v=
f, /('L
y /('L =
/('.
Y/('L = ki f1 i
k0 / I t i
S- asin
a s"il 'Y
'- + -a viny
sr(
+2
+ a
(2
where
a sin
i,
+k
(L23
where fil
-f-k IKill 1/$- siny) is small and can he
be expanded in a tourier series. After performing an
analysis with fis, tile point drag will be added sparately in
a simple way as shown in the following sections.
11(
7. FOIL DESIGN
where C'1 and ('au are given by equalions (i, 17), and
113t fisr a given foil. (r and CkI are very isccinctly
reprenlted by equatois h), (7, and ( 13) because tor a
fliarrie, An - OItn I, 2-...) with only non-tero
13
cefflicient All, and for a point drag,
I-cC
OC a c
1>14
M"l
('L
-I
1251
yI
" to
I -
I - a( '
-(s
thickneso lil te
isl
ut
ki,, k, v k+o
263
1-271
where
t
f,
k -t"
" f _ I-
(281
and
k M2 I'r
k2 F-
f2
(2)
(30)
used.
I-
Ct
M2
Then the condition of the leading edgethickness will determine the strength of the point drag, kq. Since the large
camber may induce facecavitation. the camner naturally alto
hasan upper limit.
9. FINITE CAVITY EFFECT
For a design problem with an arbitrary lift distribution
given on the foil, the solution for the supercavitating cascade
12
witha finite
cavity
length
has beenreadily
obtained by a
superposition of a simple functton of cavity length parwmeters;
onto the solution furthe infinite-cavity problem with the
same lift distribution, When
W I au - in1
(31
noticeable
Is that the curve of drag-lift ratio is almost
paedlel nafeature
very close
to the curve of noa-taill line
shock-free angle. This means,the drag originates mostly
(1,2)
where
waf)
rIs
J-
a
2
__
dl "i
u)
a"
lotg
- 3
u + iv,
133)
nJ
w,(i-O,w,(r)-=0
i
forx-.'..
t3
I'mq
0.2
5O
1
CIRCULA
ARC
0.2
CA
10C1
TWOTERM
CABER1.
TOTR
ABR
-0
-020
0.6
0.8
1.0
DC
'F
TPLATE
_.0_
1.0
0.8
CAF
-a/CLQShock-Free
0.4
0.6
0.4
SOLIDITY, c/d
0.2
Co
0
0.2
04
0.6
.0
0.6
0
1.
1.0
,%./10CL
SOLIDITY. iid
Figure 6 - Lift Coefficient C Drsgl-Uft Patio D/L of
FI y/CLO - 16(a - x)(w
Supercavltatla irclr-r
in a Cacade of Stan" Angle I Radian,
Angic, a/CL 0
Free
at Shock
is 0.1 at XI - 01 C /C 2 and the point dragsingularity0
ke/CL Versute agle Ci' tie flat plate superposed are
is positive. A
shown In Figure 8, where (d/da)(('D/C()
Q. where
at at - 0.5 is shown in Figure
s(dad)((
imilar curve
when the cavity
0 /( 1 ) is negative. in general,
is positivei..the point drag is more efficient for increasing
th edn-dethickness,
~'
~~
o the
To investigate the intfluettce of camber shaper
vriaionof cccslarsrccamer
Uft raioa
sid11wde~-l~t
is conasidered
s in FWgsse
10. The n-coordinate of the
bya factor (01l 4 bit)to (he
isVan~ed
maximum Camber
circular arc y - 44x - x3) where a tilts, - i~l)tI + hxj))
makettotaa
maximumic
thee
cabe
toe eamual
iscuse
the
is tete
Wirmximm
o mkefato
cmberequl
1,and
thesx coceinas of the maxismunm
camber is
X I-S
a . /S1-4A)/(2A) with A -3b, and R- 2(t -hb.
osfattack, and the
The lift coeMcient. the shock-freer artajei
dnagfl ratio of the defored circular arcort ahown in
C, 0
5 cu /C"
*400
0
0
a/CL "Galt'S
CL/f
ko
Ci-
10
20
30
40
FLAT PLATEANGLEOF ATTACK.ck/Ci IDEGREEI
CL
b
R CIUL
RCR RC bto
1).2
.-COORDINATE
OFMAXIMUM CAMBER
0.4
Ott
0.5
1.0
a/e
1.0
.
C,
-_________________
ICL
*0.5
-1.0
0.
0.
0.
10
1.
14
irk
andLift Distribution
Figure I2I Foll.4ianity Shape
with c/d - 0.471118
in a Supeecanllatift Cascade
-I
01231I
1.0
028
0.00.
0.
,0
.
DESIGN
SECTION4
R/r
CAVITY- LENGTH.
0A~
25
(avity-L
(01Figure
SIt,1,
0,4RiO~n
0 .6
0S
0.6Aflthough
> 0.4
0.7
Li
0.n
01
9use
0.2
0t
0.16
Cos
LEADINGEDGETHICKNESS.
I/CLlit'I 'a'iii'
1
0.26
secton
nube? cvirs"
ACKMCJLEOGEMENTS
%ml, ii'
-ii
r, upoiled b%tire N'ii 1l.1i,,
I iimari)s,,I I.,I.,miir I uirdringt 0 , Irj'
ons Pfl'tulf'i
,s lnith!siuer %asal kvhtude.
I II, aijthi' A"s, sst, I"i
prt- hisapriii,
II
I,- Il, Niufin,- M %title lir
iiti
rumi uith i tr, -i,l,,
arid tou1), Will,,,,,I B Shnpr.,
.,i Mi Jiisn Ii Wdiwi
ft I,,, lirr-ir mii %i..l,icu
REFERENCES
1. Tulin, M.P., 'Supercavitating Flows - Small
Prtiurbation Theory, " Journal of Ship Research, Vol. 7,
No. 3 (January 1964).
2. Betz. H.,and E. Peternobs, "A.4pplication of the
Theory of Free Jets,'" National Advisory Committee for
Aeronautics TN 667 (April 1932).
3. Cohen, H., and C.D. Sutherland, "Finite Cacity
Cascade
Flow."-Proceedings of the 3rd U.S. National
Congress of Applied Mechanics (1958).
Jhnsn,'he.E.Iflunceof ept of
4. Jr.
4. Jhnsn,
r., TheInflenc
y.,
of ept of
Submergence. Aspect Ratio, and Thickness of Supercacieating
Hydrofoils Operating at Zero Cavitation Number,'" Second
Symposium on Naval Hydrodynamics, Office of Naval
Research, Washington, D.C., pp. 319-366 (1958).
19. Glaueet. H.. "The Elements of Aerofoil and Airscrew Theosry."-Cambridge University Pris, London (1926).
20. Cookey. 2W., and J.W. Tukey, "AnAlgorithm for
the Machine 0caukllon of Comnplex
Foutler Ser-ies,"
Math.
Comp. 19, 90. pp. 297-301 (Apr 1965).
21. Gilbe -1 D.. Jets and Ctacl~ies.
Encyclopedia ot
P'hysics, Vol. 9, Fluid Dynamics ill, Springel'-Verlag, DeilIm.
pp. 311-445 (1960).
Abstract
A discussion of the use of boundary integralequation techniques to solve cavity and
jet flow problems is presented. The need to
model such flows as a mixed-boundary-value
problem isdemonstrated. Application of the
basic method requires four steps: a suitable
discretization of the problem, formation of
the resulting matrix of linear equations, solution of these equations, and use of a systematic shifting algorithm to improve the free
surface location. Although progress has been
made recently, examples show step four isstill
in need of further research and improvement.
To illustrate procedures, simple two-dimensional jet and cavity flows are presented.
I. Introduction
In recent years the design of high performance surface vessels and other marine vehicles
has continued to focus attention on the need
for computationally practical solutiois to
progressively more complicated jet and cavity
flow problems. Iydrodynamicists have now
sought these steady-state solutions for over a
century; to date no totally satisfactory
approach has been presented which allows a
convenient and suitably accurate fully nonlinear computation of these flows, especially
in throe dimensions.
During the past decade the use of conformal
maping, finite difference and finite element
techniques have allcontributcd substantially
to progress in understanding jet and cavity
flows. However, none of these approaches is
free of limitations or shortcomings, and the
great majority of the work has been restricted
to two dimensions. Conformal mapping Is of
course limited solely to two dimensions. From
a conceptual point of view, finite difference
and finite element approaches to jet and cavity
flows can be formulated in terms of a velocity
potential so that there is little if any difference between a two-dimensional and a threedimensional problem; the shortcomings here are
mostly practical ones.
toth finite difference .idn finite aiwmnt
techniques must fill the tire flob J-main.
The physical doeMin is always -ret ja - in
shape and often possesse, cur.
t, nadir
surfaces. Trial free-boundary techniques depend upon shifting of the free surface portion
of these boundaries as thesolution proceeds
iteratively. Finite difference methods are not
ideally suited to the treatment of boundary
conditions at non-rectangular boundaries, and
so they usually either (i)useirregular computational stars there and suffer some loss of
accuracy, or (ii)solve the problem in a velocitypotontial and stream function domain [1,2]
and tr.sform the solution back to the physical
domain, thereby sacrificing theability to prescribe the physical geometry a priori. Finite
element techniques do not have these problems
but do require an automated mesh adjustment
scheme near free surfaces, preferably a quadratic representation of the velocity potential
and possibly the use of isoparametric elements
along the boundaries.
Injet and cavity flows the most interesting
and useful infonrmtion from a solution are the
spatial location of allfree surfaces and a
knowledge of velocities and/or pressures along
the boundaries. This realization causes one to
feelthat full-domain techniques stch as the
finite element technique are ina sense computationally inefficient, for in the course of
developing the required boundary information
it literally generates reams of interior data
which isoften of minor use.
For these reasons the search for superior
computational techniques for high-speed free
surface flows continues, This presentation
suggests that a variant of the boundary integral equation method holds promise of being
such a technique. In this method allcomputations dealdirectly with domain boundaries.
For free surface flow problems the vital missingingredient inprevious Implementations of
the method was a systematic algorithn for
shifting the trial free-boundary between successive iterations; the present paper presents
two possibilities.
II.
Basic Theory
(1)
in two dimensions
(2)
G - 1/r
in three dimensions
(3)
I. Discretization
Equation 4 normally can not be solved in
closed form for o and aOs n. The usual
approach is therefore to discretize Eq. 4 to
obtain a set of N algebraic equations in the
discretized unknowns 0i and (a n), where
1 = 1,2 .... I, j = 1.2,...J and I t
N so
that the equation system is determinate. This
process involves i discretization of both the
boundary geometry and the urknowns.
3=
where
r = rpo ; the
from any arbitrarily chosel point distance
P Iii , to a point
Q
on the boundary r. By use of a limit process
in the neighborhood of P, one finds
0bp
*k
-G
dF
(4)
an 3 an)(4
11
J~drsON-1
(b) Approximation by N segments
Figure 1. Discretization olfboundary
Geomet.
Figure I depicts in two dimensions the process of approximating the true
b
g
by ddividing
i
i into N segboundary
geoetry
by
It
problem of exteri-
ments of known
(by coolce) shape -- straightl
segoentn thi
sap straiht
270
0(s)
and
quadrilaterals because adjacent segments sometimes can not be aligned to give a C continu-
11a
j'(s)
nn
j+ (0j+l-0j)(ss )/(sjl-s )
1j
at
;Aj,
an
do
a Iii n
s-s/s
(6)
l-S j
(7)
2. Equation Matrix
When Eqs. 6-7 are employed in Eq. 4, the required integrations can be completed in closed
form to yield expressions linear in the 0j'
and (a0/n)j s and involving logarithmic and
inverse tangent functions. In a higher-order
approximation the superiority of circular arcs
over parabolic segments becomes apparent here
due to greater ease of integration. In axisymmetric problems where integrals can not conveniently be evaluated in closed form, one must
use numerical integration with care to achieve
accuracy. On the other hand, when rpQ is several times the segment size it is often possible to use simplified results in place of
lengthier, exact expressions; this factor becomes most important when three-dimensional
problems are considered.
When Eq. 4 is evaluated for each of the 14
points Pi (I - 1,2....
N), the result is a set
of linear algebraic equations of the form
1t '1 *
+
,
im
Q
P
'PQ
/
write
(8)
V -'J,
(9)
Equation g is obtained from Eq. 8 by transposing to the right side the N terms known
from the boundary conditions, and qj is the
remaining vector of unknowns.
At some points on the boundary of the flow
it is desirable to represent exactly a discontinuity in ao/an; in particular, this toccurs
at a corner where one segment represen ing a
wall connects with another which has fluid flowingacross the domain boundary. To medel this
case exactly. one simply places bio nodes t
and j atop one another at the corner (they a,e
connected by a bo,,diry segment of zero length)
One can then ass
, bondary value for aoan
to fode i andi
.kij J; if only one value of
p i n Is speciticvi, cm:n * can be specified
271
W
t (n)
Vm(n)+
onTnnT
V - m(n)(a 3
(10]
l)
E
(a)Diffuser jet
C'
2--2
Figure 3. Schematic of free surface
located, by assigning nodal boundary values of
* by Integration of 34/3s - V. usually
A_
R2
----4'E
(blJet or cavity
cigure 4. Jet agd cavity fow examples
is
BEprooutlet section
section iJ.
iiiet
c,
s-dimensional;
thecndvelocity
aq01n is
5.9
withlip Vto conspverl
lower value;
caused
the velue
of 3iterations
$fn at the
verge toward a positive value. Hence, the correct value of V must lie in between, end several automated interpolative sequences gave a
final free-surface value for 3a/n at the lip
which was appcoximately -C.036 when V - 5.94.
The initial and final free surface ar plotted
in Figure 5.
2.0
Lip
.5
Final
e"
1.0 Initial
V. Pr
g.
Formulation of the boundary Integral equation method as a mixed boundary value problem
holds promise for the efficient solution of jet
and cavity flows. In several respects the
technique is still not well developed, hcwever.
Although some free surface location techniques
have been shown to work for certain kinds; of
problems, much room for both Innovation and for
improvement remains. Extension of thee techniques to the much more challenging, prac:tical
and exciting three-dimensional jet and cavity
flows remains to be accomplished, but prospects
uf eventual success appear reasonably good at
this time.
Refer
I.
273
3.*"
Shi
pp.1143-1152.
19. Street, R. L.,and Ko,P. Y.,"Numerical
methods applied to fully cavitating flows.
with emphasis on the finite element
method," Symposium on Hydrodynamics of
Ship and Offshore Propulsion Systems,
OatNorska Veritas, Oslo, Norway. March
1917.
"esfo w-iesoa
20. Larock, B. E.,*Jt rmtw-iesoa
symmtric nozzles of arbitrary shape,"
J. Fluid Mach., 37,3, 1969, pp.479-489.
DISCUSSIONS
of three papers
_BENEATH
Invited Discussion
M.P. Tulin
Hydronautics Inc.
The vastness of the literature for steady
plaii;,r
free-streamline f;ow attests to the power
of available analytical tools for the solution
of certain physical problems (especially supercavitoting flows) through their reduction to
mixed. boundary-value problemrs.
These tools
include: suitable modeling (cavity termination
and wakes,, analytic functions and mapping,
bounuary integral equations, asynptotic (i.e.
linear) approximations, and combinations
thereof. A sizeable number of important engineering problems have by now been given adequate
tionthroughi
the application of theory. I
wo; d mention especially the design of supercavitating propellers. The second paper (#18)
today, 6y Ym. isa further and very elegant
contribution to this literature, He utilizes
known tech.ques of supercavitating-foil theory:
linearizations and mappings to the "airfoil"
plane, co-inations of incidence, camber, and
"paraboli type" thickness (really a leadingedge sinliarty). He has succeeded in a very
useful sfithesis of these methods and finally
applied ,ne fast computing techniques. I was
especi,,
y pleased to see the hydrofoil-3irfoil
equival ace generalized, as these rules Imnediatel,ivcmuch insight as to the relation
thechordwise pressure distribution (at
betweef
design) and the lift/drag ratio.Yim's results,
which include free-surface (infinite Froude
number) and cascade effects, attest again to the
power of analytical methods, more so inthis case
than to the power of numerical hydrodynamic techniques.
Uieful and general as Yim's results are, I
would like to suggest a little warning in connection with their jtilization inhydrofoil or
propeller design. Inboth cases, it isoften
necessary to utilize non-linear corrections based
on theory. For foils it isalso of someimportance to take Into account the down-wash due to
gravity effects on the cavity, Further, I have
some doubts about the utility of the planar
cascade model in application to propeller design,
except to estimate cavity-section interference
during off-design uoeration--and this is the
design region where we most need better methods.
In the off-design regime non-linear effects are
a-IS
O.X?
0.4.
a-10*
0.20
0.29
610*
0.$6
0.54
*
*It
Ta
JI
Abstract
This paper describes the status of a computer
program for solvinq the full nonlinear problem
of a two-dimensional body performing steady
translation near a free surface. The method of
SolUtion Is based on distributions of simple
Rankine-type singularities on the body and on
the true location of the free surface, which
must be obtained by iteration. The key is the
iterative algorithm for determining free surface
shape. It must account for themutual influence
of various portions of the surface. Various
stages of development of the procedure are des.cribed, including the final successful technique based on a higher-order singularity procedure. Comparisons are presented of freesurface shapes calculated by the present method
with those calculated by a previous finite-difference solution for a submergoO point vortex.
For the higher-order solution, agreement is
quite good. Futu" directions of the work are
indicated.
Introduction
The problem of Interest is that of the
steady translation of a body in the presence
of a free surface. 1he fluid below the free
surface is inviscid and incompressible, and the
flow is irrotational so that it is a potential
flow governed by Laplace's equation. The fluid
pressure is constant all over the free surface,
In three-dimensions this probam finds Its chief
applicatin in the calculation of wave resistance both for surface ships (the surfacepiercing case) and for undersea vehicles (the
submerged case). In two dimensions the flow
about hydrofoils Is chief application. Although
the problem of main practical Interest is the
three-dimensional one, becauso of its very formidable nature, the present aIffu.his been
devoted to the two-dimensional problem, Whre
the only solution techntques considerejd are
those with direct three-dimensional analogies.
or
V*"'t-ztg
where
(2)
To solve this problem by a surface simngjlirIty approach, the body surface S is covered
1
l.
' W
T
-
-*"--fi
-.-
''
Lt
normal and
(ifany). Then thetotal
itsimiage
tangential components of velocity at (ii~
are
by the present method of this report can be comnpared with theirs to yield anessential quantitative test of accuracy.
The essentials of the point vortex problemEA
are illustrated inFigure 2. The vortex which
hasa strength K. is located at thepoint
xi;0, y --h. Here K -r/2w where r is the
cirulation. The problem has been addressed bothE
with and without an image vortex of equal and
opposite strength at the point x - 0, y - +h,Vi
and itwas encouraging to note that the convergent algorithms yielded exactly the sam results
in both cases. Durina most of the work to date,
an image vortex was included. All cases follow
references 2 and 3 in using a freestream velocity
Ui of 10 feet per second and a submergence depth
h of 4.5 feet. Various vortex strengths K are
considered, which lead to various wave heights.0()
During most of the present effort only reference
2, not reference 3,was available. In tnispaper
allcoparisons are for the twovortex strengths
of reference 2, K - +1.15 and K - -1.4. It
should also be noted that the results of references 2 and 3 have been ootained for a finite
fluid depth of 9.5 feet, so that perfect aqreewent with the infinite-deoth results of the
present method cannot be expected.
The free surface Is represented by singularity distributions from MI or x ( see below)
to Xlg(Figure 2). From x~to Nthe
velocity is required to sati sfy the free-surface
condition of constant pressure and the shape is
allowed to vary to produce a condition of zero
normal velocity. The physical variables are
u. h, and K. The "numerical variables' are
length ax x +-I
xg. , xNz. and the elemenit
-iwhich is used to define the free surfico.
spacing is
a
constoat
Inill cases presented,
usad. and Ax is a single number.
VNI 2
Usit +
11"1jr
-u5n~+v
11
UCoa+V
u cso
+
v
+i
iivi
(3
(3
(4
(4
-2Qy
Vi(ii)
(6)
surface shape.
Local Algorithms
The simplest algorithm computes the ciange of
slope angle 6'Oi of each element as
6o0 . tan'(V1 /Vri
NiT
These are added to the -I to obtain new 1
which in turn are used to calculate new y
successively, beginning with som fixed upstream
value,, The t# enieclua Ioni eetd
This agorithm converges fur theclsia
iitvorse problem of potential flow, in which the
prescrie value of tangential velocity at each
(ii.ji)is independent of location, but It
"___7
\J
When the free-surface shape is altered, firstorder changes in the velocity components (3) and
(4) are
Ni
LAijavj
+
NVi
(eVi
(8)
--,(VvTi) ii
A s gFig.
(9)
i Free surface shapes for K-1,4.
First-order solution, no flat. Ax - 1.
Ejj
(0)
(11)
[Vi(ji)]vyi
(12)
261
I
,
'
'
,.,.
-- - ,, .,:,
::
...
Q,,,.
,
,
,,,.
, /\\
It turns out that the above-described sensitivity of the results to thenumrical parameters
is due to a nonuniqueness arising from Insufficiant numerical precision in the first-order formulation of the surface-singularity technique.
This can be illustrated by a saI*le calculation.
An original shape was selected that is flat from
x . -30 to x - -10 and that has a wave-like
shape for x -10. It Is sho" as a nolad curve
in Figura 9. Flow about this shape in the
oresence of a uniform onset flow parallel to the
-.. ...,,. . .
Fig.6 Free surfve shapes for K * -1.4.
First-order solution. ux 1.
1
with those of reference 2, but the wave lengths
are somewhat too lonq. Unfortunately, this
Mi
i
...-
wi
t
, . m
MTIMU
Sf, n
.......n
*flICACMLtM.
-
M,,
U,-,
, ,
kMII
w11
f b
K S+1.15.
AX
1.
,,,
If
In
"
n ,- ,,,
,
,,,
..... "
UO *,So.
-,LCMC,''U.
,
above
the results of tests of the iterative algorithms
may be sunmaized as follows:
Higher-Order Procedure
When the higher-order surface singularity
procedure of reference 4 is incorporated Into
the above-described global iteration procedure
with initial flat, the result is a method that
oppears to be quite successful in predicting
free-surface shapes. Moreover, it appears to
be stable with respect to the numerical parameters. Calculated results for vortex strengths
of +1.15and -1.4are shown in Figures 10 and
11,respectively. rheagree:Bent
of thewave
shapes obtained by the present method with
those from reference ;', is essentially exact
when due account is taken of the fact that the
shapes from reference 2 have been calculated
for a finite depth by a numerical procedure.
The curve of the present method in Figure 10
represents three graphically indistinguishable
263
---
I0 and 11 that the calculated curves have erroneous bumps or bulges immediately downstream of
x = -10,the end of the flat. These are due to
the fact that the numerical differentiation procedures in the higher-order method are somewhat
unstable at the junction of the flat and the
curved portion of the free surface. A straightforward modification should remove the instability and smooth the bulges. Evidently, this flaw
in the method does not have a large effect on the
calculated free surface for these vortex strengths,
but it might at larger strengths.
Some improvement of the iteration procedure
seerfdevirable and quite possible. FPIiro 12
shuws Oev free surface shape computed by the
present method in the first iteratimn and in the
final sixth iteration for which the normal velocity is everywhere less than 0.1%of freestream
'
'
von Kerczek, C.H. and Salvesen, R.: NumericalSolutions of Two-Dimensional Nonlinear Wave
Problems. Presented at the 10th ONP Symposium
on Naval Hydrodynamics at M.I.T., June 1174.
"'I("'*2.
4. Hess, J.L.: Hiqher-Order Numerical Solution of the Integral Equation for the TwoDimensional Neumann Problem. Computer Methods in
Applied Mechanics and Eigineering, Vol.2, No. 1,
February 1973.
264
Abstract
A numerical procedure is presented for solving
free surface flow problems. The boundary value
problem in terms of Laplace's equation subject
to non-linear free-surface boonrlary conditions
is considered to be composed of two fundamental
subproblems.
The first problem is defined on a fluid domain
with a fixed boundary and is known as the Neumann
problem whereas the second problem consists of
the determination of an improved position of
this boundary. The solution is obteined by a
process of successive iteration on both subproblems. In view of the intended extension to
the three-dimensional case, the Neumann problem
is solved by a procelure using finite elements
in a dimension lowered by one as a result of
an approximation by splines.
Up to nov the correction of the free surface
has been developed in a two-dimensional version.
The procedure was applied to the two-dimensional
steady state problem of a running strew with
a flat bottom. Then, somewhere upstream a
disturbance of the free surface is introdaced
in order to calcalate the effect in downstream
direction. Besides this, numerical results were
obtained for a running stroam, upstream undisturbed, with a bump on the botts.
I. Introduction
Contrary to methods using singularities * at
the boundary of the flow region, the solution
of the problem of determining the wave resistance of a moving obstacle partially immersed
in the fluid in based on a direct numerical
method. The advantage of this approach arises
when the method is extended to cover the case
where the non-linear effects resulting from the
dynamic free-surface bundary condition is taken
into account and atteaps are made to solve the
three-dimensional problem. The direct approach
appears from the simplicity of the procedure
to solve basically the Neumann problem on the
one hand, and the problem of iproving the
free-surface elevation on the other hand. The
difficulty, however,is hether convergence can
be achieved when both subprotleu are solved in
an Iterative way. Starting to solve the Neumann
problem which is formulated on a fluid domsn
of a known shape, it is evident that the results
can be used to calculate a new free-surface
elevation from the dynamic free-surface boundary condition. The kinematic free-surface
Srot
, 2,
3, 4, 5
Ay
--
Y(2.1)
no
Fig. 3.1
- 4 +
y
x = 0
xn
(2.2a)
The known free sjrface elevation y = i 0
corresponds to x 3 - 0 and toe bottom y .-E(X)
corresponds to x = -I. Using the metric tensor
g.., Laplace's equation is described
by the equatign:
_{x2
(12
(2. :b)
2 =quantities
11 2
a- (ij 2-4 ) = 0
x
3x
an
where
and
iU
at hI
h
det igi
(,.3)
an
(3.1)
ax
ax'
'h 0
,32(O as
u_
g n
Il) X
anSInstead
ax
U0
y - n*(x) we look for a new position
free surface determined by
of th
y - r(x)
Ill. Comrutation of the free-surface elevation
3')
as + , 33((-(A
_ 3
ax
, (x)
X(a)
or
x
ar
Z(x
dO3
d
ii
d
or
*r
. 22
dx
dx
d
dx
or r - r 0 *
(n can be approxiM
leads to the kinematic free-surfat
206
)
condition
33 at
3
3x
3a
2
ax
22 at
3
d
2
dx
23 3
ax 3
and
and
q .
(.b
I_
dx3
3cosh
x
wer
-
x
-0
inwhere
a
2
2
3X
,,23
2
8 is equal
to the mean value of 0 in
2,
an interval ix
Furthermore, the boundary condition at the
bottom xj - -1
2
d 2h.
d3)2L- - 2 isi
h
dx
3a)
wheru
(3.5)
2
2
a. 8(x )f.i 0
22
(dx) 2
(U3 - q') - g no
)-t-33 G4 ?J-
3
him)
3s
.'
and
ax22
(ax3)
xx,x ) = { f.)x)
21x3 so 1
2 cc __ cx '
3m
ax
(x
(3.2b)
Now, the potential $ is considered to be
composed of a contribution of the potential
determined by the solution of thi preceding
Neumann problem and a perturbation potential
with the assumed property:
2)
2 2
(x )
or
30ax3
is atisfied,
3m m
,xs)
s~a.
by the
5i
and
'
o(m ) 31 3_
)x
+ :3
3x
#_3
3
3x
ax2
)~
(-)
(1.)
-I
267
2
0
sg tanh
a H -0
'3
2
q
If
(3.10)
t
tanh ai
- c, A
r.clation.
The value of 0 being known, the solution can
ax +
solution for
)=aecos
we find the
dx
at the eid of the interval. They are the initial
the
next
interval
and the calculavalues for
tion of C can be started again. This process
is sequentially repeated until the end of the
region in reached,
As mentioned earlier, the value of 0 in an
from expression (3.4).
is calculated
interval
of the approximated solution (3.5)
Substitution
(3.12)
F = c cos a x
Continuing the process we find that above
first
result represents the solution in the free
order approximation for the complete
surface. Subsequently, the Neuman problem is
solved for the new fluid domain bounded by a
free surface nl which is equal to above value C.
With respect tR the velocity field obtained
this value is replaced by a value n* in
agreement with the dynamical free surface
ail x.
1
where
Po23
aelevation
+ 1
3 _
ax
5 0t2nt +
023
i 0of
and
401, +
ax2
(033
an
a h
2 2
a G
The value or 0 is calculated from the requirement talt the rorm P2 + Qcal
e
fr, minimum
23
constnt
imave
In the cae when
f 0
0 and G"
which occurs at the firs and p - and Q can
be simplified to:
22
P-2tan
dx
20
33
2
tenh ai (O - *Go")
(3.1)
....
(3.13)
((22
interval.
"
an
x3
x3 .x,1
resulting in
of the third degree.
'Sheapproximation (4.1) is used to produce a
set of ordinary differential equations,
following from the requirement that Laplace's
ion isor.ly satinfied on the curves
x
Arrang ig this system, the result of
J
method is represented in a set
the collo'eation
of n ordinary differential equations of the
form:
Xerat
x .o
4
3
x=-
CU
0
Fig. 4.1
d (G 2
2.((
dx'
1 n
~--. 3.3
-- n
~(J-------~ -x
-
-- ------
whr
whr
d
2i '
(4.2)
+ 3
x
()
-x
1
1I
AJ
ij,.:
Ij n
-2 -I
1
7,
'
1'1
1
combined with n-2 relations, actually resulting from the requirement of the continuity 3
of the second derivat~ve
with respect to x
3
at the intercurves; a = X3 (jC,(I)n-II.
3
j+1
3x.
and
quantitis
-unknownis
at their boundaries. It
nl
is
V. Numerical results
expressed by
(J
'5
In this
3
3
~(1
(su
(J)
1 2X3Q)
I
Firat,
iare presroted for two different exasaplee.
nu~merical
metho)dwast rheciod inttheccase
ithe
+.)ji
of
runn
trew
o
local disturbance at the free ltb
surface
represented by some de-iation with retipectto
k.0
3
(.
20
0.02
. ...
-
-+'t
-L
I_
-A'
ii
+,
--
-~-
--
,,.
0=
0j
V
0.33
_.
0.2
i ig.
Fig. 5.1
The method turns out to be stable producing
results characterised by an unchangable
difference between the free surface elevation
aluen
during the last"
and its corrected
c
5.2
---
--
o0.
1.75
I
V!.
Fig. 5.3
Conclusions
290
ABSTRACT
Nusineical solilsIouIs
Of tIre nonlris-ar tproblem of twrodinrensional f-inite-depth piotential flow past a fined pressure
distribuion on a free surface are given. lice niinerical
imertiod approsimais.s ihe enact piroblern by finite differences
thle nxrverical sOiriitions presented hcrc .re-for a rangefsif
values of the Fronde number based on the length characterizitugthe free-surface pressuredivtiiutin. Thre numrerical
solutions ate ctompared ixsfar atspossible with first-., seconud-,
and tlrird-order pertiebatron-t heory solutions andI it is ftiris
that the second-order perturbation theory can give accurat,
solutions of thns problemi for the iange cif Friutde tnnmbers
iivostigated. I lie details itt the free-surface deforination
s:-aused
by lte suface-pressure distribution arc-presented ill
graphs. these graphs illustrate bele
efcts Of rthenonirvneatit c-soft the piuohicr on the description osf the wasc-making
action ruf tIIresurfact-pressure ilisirihuirion.
NOMENCLATURE
C
horwwavesaregexerated by
pocssible abouts
IIi InIiatiosuis
C Ik
10 reIPOIgIlit
C~~~~~~~~~~~~
Ilippglresteclfcitobtacles
It
vfj Frucnumertotted
* I) f~T Frirideirisribe
I. effective length (if press/ire itrtnitioti letI-r
Ylitni (7) ritFigure I -i
dirrierusori
Ii
gas atrural
It
It
Ii a twur-dimrensiounal
iram
water deptht
ft
trernrI
seleratrisi
nieth sius
V- lengthsIstetited Ini tigrire I arid in,I 'ultonrr 17;
I',, xi
it
fivee-uraeled,
c
rn
p,
A
A5 s~
waveleert~
stream tsninn
tnfudsioiir;
In thusprapwrwiepresert t iunrr at s-loi
.I-- tire
rtnrtrrvea i role iuf r wv-drrnsensrtni rinite-depthr posterntial
29
ti
It lg h rlto t
iili 11
pivon1 l lis
pilo ir.10O ilslltlilt'
I vrtitoine -. itl
Icn,1j't'ollt'l
Iii,
p, Ii
imviltric
u
lic
,il
i i
tl i'
tK'llit'it'ni
1:
i'
mIt ptt
l t t1rill Ilitt t
i ltrtirt
ot i
'.JLC
to F
m
tislil
n
.w
ifF
lollfiln
C!i
in
Ifl
nricir
sHomrI-
s u e
~,I F.W ~
of
thel Pcrblem
obaie
ho.t'i
i
).-
li ti Ill, ntIl
~ilitc,
,-w
[ti
ciii s o : psil
diet10iir
otlnl
ici=c
fi il I 1t ttli,
'1I F.11
i trl uI w r " ( )
Ii
-rlnie
itOl
" 5'
"
Ii
t1tt' If ''
I!-cll
l
,
I5lie
th
141
h~ ~
~~ ~ i, ~ m ~ae.n ~le iilulc
~trhcict
Figure~~~~~~~~~~~~~o
Ih rcSracPOFI
0
.1ti Ill011~l"'Ilcir
\k trm lt Iikpolle
jilt't
il i c
tIillitl
iiorllpi linlt''illt'l'ilt
IF iKtr'iii
Ig lii iirMethod
,t't''Flll lltrilU1t'1
tg
it
it
i l
Mid.
w t"
liii
ilttt'ilt'll
lllll'tl
thi
Isigi
=c\ji
rc-,il
11111
ucer alM to
illir
tiretili
il!i~n
Ii_
'_
trw
lilt' tthect'itlt
e
tilirililior
itlil litT
t11
.lii.11
\IF
cin
itioFl
All
millio
%ril
i'W lj
kI
L-elocd
hs
lvir
i''41h
. F rto
nlni-ipt .- fo
rnvoulto
liill
Ikli
lt
o td
~~l',1"'ie
-\,c
is
(IlWe'rF
I
111
'rcmi
l'ltcil
jelil tilr t he
l ll
~~
itio
lt1.
or
tlicet
iii
tluntO
t l e
~i
t thtt-t'in I
111
ito o1 1 I
r ill
Ii ll
Aritittli
p.r
ite i'
+Il
'1111
tlr o i
i- i iiR leene
l~lliriie
0 1ti,
1.
n~ C
inth
n a211
grid whic
ii. I I I I c o
hical it
i~j- slp2i93i
11
it
lt
Sleti0lis
ltqt = -4
/c
-f
now retpeated.
1!5)
&'l
pol I Illt
Of
00i1
77(
+ ) 091
VWg,4n
ite
.
,ii
2I
..
(8
ttliotis
ix giver tbp
dx
aid
i~xl
do .
i
IR=-
Il
Ihe initIcg
ral In ilt91is over ttie lenItgIt(IIf lie p1ressurI
e distni-I
luion atid is evailiiated tos~rgIratWieeoitutadra tiires. thle
f'ree-suruace slope I dhtids I is coimputeutusing lir'it-iirder
10)
VI- 11
1I2
cetitral differeiice,.
where a, is the first-order Par-dnwnstrean waxe amplitude,
This implies that the wavelenigth, according to the thirdorder theory, is
Perturbation Method
It is Issumei thai~t thlt stirLac liressure idistibution isa
weak distitrber vii that tile pertuirbiatiois ibtit le uiiifoirmi
[liiw are eserywhiere stmail.and that ithe streami
i-ell las
lite fiilloiwiiig exniion initernti t 1 perturbiation
X=
inf
'n
tpaaneteri
lW
R,-
are!
where I'l - troWtlu and lik5 with t - 2 -.
snlions 0i - 1 anid I?("-It
Paintonsiif tilehiiwerniriler
Ire lii. Irliaions
.1ASand L174and where Li wi is
socittn are giveit
deptht thle
Fourittinite
tile wane oniie
integral 1171 1i. 60i1ith
tbylie conilitiion
I lie liiieiri/ed
and Itwoi higher-order wave resistanes
for the lurtsisrt distrthiiioit lane tet obtained tip
numterical integratiotiti
I)C
( 121
ui + u(" + ka,
pt i
its
,n
1 2. j .. 114f
I
.qt 1. 2.l
dx,,,,r
n1X.nl.
2i
1224)
'
__________
2p (I -ylitch
s~n)! ibfNh~l~eI 1iir
1211
Ut 1v
ii
k.
0 DI
where
Suikupt.
aretused
It, mniiate
thata quantityit w'inci within therorderof tbnWtlrblttui thilyas isenst
by inia
index.
(1)
2.0
1.5
/i-
/
a1
a-
/NO.43
'A.
01.
0 .5I
NO.3 2
I
UNO
0.50
0.40
0.30
0.80
0.70
it
I, ()
co k, x dx
poI I in k,,x dx
p1in
~~,e
It
roni ho ottaionwy
arrd,,
oniive li. ootof
andosiive
,, s t.,
ootof
ral he euaton
irh k,, 1) - 0.
I- Ptu
Rautts andDiseussitn
We first examine the linear-thetury waneresistance oif
free-%urfaceprressure dirlhutloni of tile turin liven by (7
it Figure 1. Figure 3 shws tihe infisite-de~pit
shorwn
and
*linear-thory
wave-reisnance coetficienit (l. - vernnsFroude:
number IF forr the four pressure disitrbtions shown iNns. 1
truh4) "1lepesrnitiutosI-4alhv
tte
I4 asihoan
throuh
rrestate I. iintrbutiors
characteristic lengthr1. srid corrrsprord to ratios of It,R,2
1, 1/4, undt It. reirectinely: I P, and q, are defined in
Eqation (7) and in Figure I). ntrustfIere
Wenorte that thCgaexitpak
at'a
e arte soat
sizs udintathe effercslo itnlin
mneste
ota
th puroslie. rf
nualvefo
itistehoisnt
av
ftitieriery
thgrp
ae
ftelna-wr
hw h rp
Fgr
reqnitaric coefficient CI versus the Frxiide number 1: (atid
the wanelength depth ratio. , -~ID) for presaure dixribtition
No. 2 anti for tinite deptti D. lIt th0sgraph. the depth irf
the water in cinistarit arid erinua'tr rnhail tire pirciniire
distrilvitior lenthi 1. '1 he isoirrt in Figure 4 are the
2.0
3.0
25
A,=o
1. -
.
0.50.U
0.3
IL
I
0.4
0.5
.,
0.6
0.30
1.6
2.0
2.0
WAVELENGTHDEPTtI RATIO, A1
'
0.40
FRaUDE NUMBER, F
1.0
=2.00
0
- Cp 0.64 NUMERICAL
RESULTS
a C 0.96
rI-2C,=126
1.2
l./
2.
a
0.50
0.60
FROUDENUMBER F
4.0
3.0
1,D
Figure4 - Wasve-Resistance
Coefcient versusFroude
Nmlselr ft=. Various Valuesof Era sure Co(efficient
C'p,
Comparedto Fuite-l)epth LiuearTheory
20
1.5
LENGTH.DEPTH RATIO, LID
1.0
It is ot essentialprac.tcadinterest to kno~wif"
highe'rorder pertnrhlation theory for the twrdimernsionat waveresislanc'e problem acco'ratetypredicts the nonlinear
variations of ('with ('p,
as shownin Fignres4 and 5, The
reults resented ilnFigne Ii gist'some indication that
secont- andJthird-order pertarhalionl theory nay predict the
waveresistance coefficient C'fairly accerately. In t*his
ligate thle ratios taf thle wave-iesitance 'oef'ficients ('C 1 ,
Ve~tls the ptrssnre coe,.ffic.ient r'pat nariotts valuesof
Frotide number,F are compared. The open symhols denote
the numeric'altinile'deth resultInand thle sotlidand dash,_d
curves demotethe sec~ond-and third-order infinite-depth
iterturbation-theory re~sults. The value of the ratio A)I/D
is less tttan 2 f'or aSlof the numerical resultsshown in
Figure: It; hncne this is almost ettuinalent to infinite depth as
far as the desc:ription
of the wavesis c:oncerned.However.
enaminatmonl of the nutmerical resultsin Figure6 showsthat
the characteristic lenglthI. is tot lrge relatine to the depth
I) for nonlinear bottom effects on the wane-makingl action-of the pressure distribution to be neligJible.
on can estintate (ro'ughly)I
the wave-reslastance
cefficient for intinite depth, LID * 0, at fined valuesof"
Froude uuntber F and pressure
coefficient ('pby eatrupolshugi to L,/D=0 the finlte'depth itnmericul data given In
F'igureIs. A hand-ahetched
eatrapolatlon of the finite-depths
values of C/C'1= Is presentedin F~igure
7 for F * 0.461 and
0.357 andit indl'aten that theenac:tInfinite-depth values of
wave-renliatance coef'fic~ent
C"wIll he either closeto the third'
order perturbation theory vlues U,C!or atl
lent between
the .second and third order valuesa
o(. Plgtarsa6 and? s7eem
to indicate thatfor F 0..S7,the thlrd-oeder theory fal=
For tite larger valuesof C9 . Weconcludnthat perturbation
theory can be enpectedto predict fitly well thewane
resistanc'e
of twra-dlmenslonal
pressure distributions, at least
for Frotide nutubem F >fO,.t. At high speedsIF > 0,55)
.5;)
'
SECOND-ORDER
THEORY L/D-01.
THEORY j-1
THR-ORDER
THIR
--
1.6
] 0 -0
F 0.451
NJ9IL
r.15)M~O
LD1.751A4
~*~0JClR
00
1.2
0
cc
D-.3
LID 1.31
-Z
1.
1.5
2.0
0.5
1.0
RATIO, L/D
LENGTH-DEPTH
0.
256
F- 0.357
36
1.
.6
51.2
Lu0.4
o
.8-
:TER
METHOD0 a'UMWRICAL
a 0 0
U~CONO-OROE5
THORY
0 0
00
0
2.5
Lo
1.5
2.0
0.5
RATIO, L/D
LENGTH-DEPTH
0.66
leA
.40
I&
0.21
1
1.5
1.0
0.5
PRESSURE
COEFFICIENT. Cp
A L/D - 3.32
2.2 -0 L/D -2.761
O L/D -2.00
8 2.0
(23). These values show that Ithedifferences besween finitedepth linear theory and infinite-depth linear theory are no
more than t 10 pe,rcent for Ihe easespresented in Figure 6.
whereas the differene,~ between the finite-depth nonlinear
numerical results and the infinite-depth perturbation-theory
results are Much larger. For example. tfle F -0.302 and
A
-0
1I.8
t,
W
= 0 have
I
F -0.302
0.
1.
4W
1.4
~the
1.2
will be Eivea.
~0
1.6
1.0
COEFFICIENT.
PRESSURE
6 Wae rawPresurethe
Reistnce
Figue
Coefficient for Various Freoude Numbers
Infreesurfceenationsbetweetnflnlteath andtllta-stepth
*Fut linsaartheory,
lthadifference
wisent
bathLID andX1.ee/D arels than2
ialessthanI perenst
97-
-.
NUMERICAL
LINEAR THEORY.LID
I
0.02
0.461
- I.00
0.04-F
0.04-
*0
-L
.0.02
-0.04-'
S0.02F 0.300
LI
- 1.3-
s.-0.02-
F- 0357
.00
L
'
4.02
.0.04[
.3.0
.2.5
.2.0
.1.5
10
HOISIZONTAL
DISTANCE,
x/L
.0.6
206
r:i
A particularly interesting case with strong nonlinear effects
is shown in Figure 9. Here L/D = 2.33 and F = 0.357.
Free-surface elevations are shown for two pressur cofficients. C. = 0.80 and 1.6O. Note that tle doubling of the
pressure coefficients from ( = 0.8 to ( = 1.6 results in
only a very slight increase at"the wave ieight land waveresistance coefficient C) although tie amplitude of the iocal
disturbance is almost exactly doubled. I has. we we that
the main effects of the nonlinearities of the problem are to
change the phasing (increase the distance between) the
front and rear portions of the (cal disturbance.
An interesting phenomenon connected with the nonlinear variations of the free-surface profile with increases in
coefficient (' is more thoroughly illustrated in
pressure
Figure I0.
lere we slsow the variation of the waveresistance coefficient (' and the actual wave resistance R
(made dimensionless by the hydostatic head pgI ) with
pressure coefficient (p. Note that as (' increases towards
.,. the actu'i wave resistance R increasesto a maximum at
Cp = 1.4 and de(rcusesagain with a further increase in ('p.
0,03
/
0.01
w
W
e ---
.001
4'
\istributions
..On
--
S.006
2.5
-w
--
Cp- 1.60
.w=
-1.5
.1.0
-0.5
x/L
HORIZONTAL DISTANCE,
.2.0
Ac
1.4-
,0
RE
NC
IST
- 0,00
a c
- 1.20
~A
WAVERESISTANCE
NMEOFHWAVEERESISTANCEN
PRESSIRE COEFFICIENT Cp
Figure tO - Nstndtmenatomat Wane Rentlaanee
and
Wave-Reabstanee
oeffkclent sessurn
049
Prrs ar C'oeffient toe F - O.357
erutC
Figre
I-aveRiiisant
Figre10
Nndfrwisonal
ad
PaveReisanc
R oelicew
eWnrecaopime
AcknowldgYl
This work was supported by the Numerical Naval
Hydrodynamics Program at the David W. Taylor Naval Ship
and Devetopmient Centr. This Program is join ty
Reseatrch
.sposored by the DTNSRDC and the Office of Naval
Research.
REFERENCES
"Numerical
IlI von Keeczek, C. and N. Salv~sen.
Solutions of Two.Dimnrnsioistl Nonlinear WaveProblems.
Office of Naval Research Tenth Symposium on Naval
Hydrodynamics. Massachusetts Institute of Technology.
Cambridge. Mass.,pp. 649- 666 (1974).
I3
DISCUSSIONS
of three papers
Invited Discussion
T. Francis Ogilvie
University of Michigan
Before discussing the three papers at hand,
I wish tomake a general comment about the subject of this meeting: Numerical Ship Hydrodynamics.
301
be able to handle real problems? It is encouragIng that these authors do treat the full nonlinear free-surface conditions, and I do not mean
to imply that this is easy. But where will it
lead?
Discussion
y5-yilsSvesen
It is difficult in most cases to determine
the accuracy of numerical nonlinear solutions
of body-wave problems. First of all, there
exist very few published nonlinear solutions and
very often the published data are for conditions
slightly different from those modeled in our
own coiputer methods. For example, von Kerczek
and I were forced to make comparison% between
nonlinear numerical results for finite-depth flow
and perturbation results for infnTfe--de-'flow.
The reasons for this are that 'ahlgher-or er
perturbation theory for finite depth has not
been developed and that our numierical
method was
302
Author's Rep1
by A.J.Hermans and C. Korving
to discussion by T. Francis Ogilvie
The remark of Ogilvie about the radiation
condition suggests that in performing analytical
studies for similar problems a different radiation condition has to be imposed.
"T.-,
Z
*
-o0
J
4
it
-0o.
.0.08
NUMRIAL
NU
IC AL
IE
-010 -
INFINITE
NFINET
\ *
FINIT
I
EXPERIMINT
*DEEP WATER,hIXu 2.0
THIHO-ORDIN THEORY
-0.12
0
-I-TH0KDOERTHEORY
\
\\
O.
014
O.K
O.K
0,10
DIMENSIONLESS WAVE HEIGHT, HI/X
Author's
byoh iiRepl
. ess
to discussion by T. Francis Ogil'!Ie
0.12
Figure
I
303
30
Abstract
In this paper two time-dependent numerical
schems for solving steady and unsteady potential flows for nonlinear free surface problems
are presented. In one scheme, the finite
element method is used to make the field calculatrionof the velocity potential and the finite
difference method is used for the ttee evolution. The feasibility of this scheme has been
demonstrated by numerical golutions obtained
for the two-dimensional problems of the pressure distribution and the submerged body.
In
the other scheme, a finite difference method
that couples an explicit, single stage,
second order time integration scheme with the
solution of the Laplace equation for the velocity potential is used. The feasibility of
this scheme has been demonstrated by numerical
solutions obtained for a two-dimensicnal
pressure distribution problem and a threedimensional accelerating strut problem,
1. Introduction
Free surface wave problems are characterized
by complexities in flow geometry, flow features
and boundary conditions. The flow has an unknown free surface end it is propegative and
transient. The boundary condition at the free
surface is of a mixed, parabolic type and it
contains highly nonlinear terms. In the
steady state, there also exists a radiation
boundary condition, since the waves, once
generated, propagate dowstream. These complexities have lee to several computational
difficulties: accuratel) accommodating the free
surface geometry, satisfying the boundary condition uniformly over the free surface, and
treating the radiation boundary condition,
Computational methods have been developed to
deal with some of these difficulties and have
been used to solve several free surface problow. These methods have advantages as wel
as problms in their Implmentations. Some of
these advantages and probsms furnish possibly
useful guidance for further development of
computational echase.
Van Krcsek and galvesen [1.2] adopted a
marching technique to eliminate the necessity
eaeer5
of the radiation condition by guessing and correcting the free surface for steady, twodimensional flow. Their dovnstreat. closure
condition is simply an extrapolation of the
streamlines, but it may be useful in evaluating the nonlinear effects. Steady stare solutions can also be attained by using a transient
approach. The works by Chan and Hirt [3) and
Haussling and Van Eseltine L4,51 showed that a
local steady state solution is obtained relatively soon after a sudden start. It seems
that one of the most general methods for free
surface probLems is MAC[6-10 or its modified
,erston SU,2AC [11-13], which is an Eulerian
method with Lagrangian u"
ling of the free
surface geometry. However, aestions still remain concerning overall offic uncy and applications to problems with complex geometry.
Lagrangisn methods [14-181 look good for simulating free surface flows in confined regions,
but are more complex for problems of flow past
obstacles. Furthermore, the Lagrangian mean
would be distorted so severely s to produce
serious questions of accuracy.
Several investigators [19-26] have succeeded in applying the finite element method to
several free surface problems. However, there
are many unsolved problems concerning acturacy
and covvergence of the finite elment solutione, especially for nonlinear free surface
problms with far-field radiation conditions.
Chan and Mai [25] and sit and Yeung [26] used
the known radiation condition at a suitable
distance and atched It to their interior
numerical solution at the truncation boundary
by u.Ing an sigenfunction expansion.
Further application of methods developed to
find accurate solutions of aore compies probIoea depends on the success of future efforts
to alleviate the computational difficulties
associated with each of these methods. We
shall describe in thiv ,aper our research
effort in the development of computational
methods for nonlinear free surface problms.
The focal point of our study is the nonlinear problm for a disturbance in uniform or
accelerated motion on or near a free surface.
Our objactive is to develop numerical nethode
for solving steady and unsteady potentLal flows
for free surface problems.
g80--S _0Oo
J4:
# - x
-
c - (a - b)=
where a and b are the semi-major and semi-vino
axes of the base ellipse respectively. The
base ellipse is represented by b- nJ(sab)c .
The metric or scale factor h assoclated with
the transformation from the Cartesian to the
elliptic cylindrical system may be expressed
by
h2 _ c2(ainh28 + stn29).
The problem considered is defined as follows:
in f.
" # -
on y
0
2
2
1y-(+TI 0)/h 2
44
Fr
(6)
2
(04)
on a
(2)
#644 8h2#yy
#t
()
P0
"L
II Fonsoderm
olat ofthe Proe
We consider the Potential flow produced by
a disturbance moving forward on or near a free
surface. We areume that there are no secondary
motions such as pitch. roll. heave, sway, or
surge snd Lhat there are no ambient waves.
The flow is governed by the potential equation,
free surface boundary conditions and conditions
at other boundaries. The free surface is
characterized by two distinct conditions,
kinematic and dynamic couditions.
and
on solid boundary.
an
on cut-off boundaries.
0't
0
(3)
U(oh A ,)(co o)
1x.
22
x Y
o4y - yi.
i n.
ll,
Il
y -
.
Fr 2
rC
finite
Kl owt
1Z i
retho
30
For the solution rf the free surface prublemo presented in this paper, we use an
iterative scheme to solve the matri;. equs.ins resulting from the finite element formulation and a mesh system which is generated
nusmtrically
in an optimas way. we shall
describe here th iterative finite element
method, the rethoJ we use to generate the
mesh system, .rs. their significances.
In the variational finite element method,
the form of the unknown solution in assumed in
terms of known (trial) functions with unknown
adjustable parameters. The sasumed trial
functions are defined in each element with
continuity requirements across the boundar'es.
'he fin-te element solution is to obtain the
combination of trial functions that extremize
a given functional for each of the elements
in the computational domain. The solution of
the Laplace equation over an element is
approximated so that the approximate solution
minimizes the functional
] -e[
_ ,(vo)
dV
[")] [k 8
[K]
(7)
.,0
(9)
101
which yields
l(-Jf0I)
(10)
(U
J - {S
i(14)
(8)
(
[f(xyz)]i
(13)
Instead of constructing the system stiffness matrix and rolving the reaulting matrix
equation, the solution precedure is to construct the sector stiffness ritrl.xfor each
kC]
where N is the number of elements in that seccoi. The matrix equation in a sector is
to
re
in
that element, thus satisfying the Laplace
equation, For the boundary eln t, bot
the non-Dirichle boundary conditon and the
governing equation should be satisfied.
Ce
E
e-l
(12)
307
algorithms.
However, when the triangular mesh
is used, the der,vatives have comparably larger
errors.
In this case, the spatiA). derivatives
are obtained from polynomial approximations
after the iterative solutions converge at each
time-step.
The successive overrelaxation method is used
to solve the Laplace equation iteratively with
the specified values of * on the given
boundary 1). The updating of 1 and P* is carried
out by the predictor-corrector method after the
iterative solutions converge within a required
limit. The predictor-corrector method has less
stringent coeitions for stability and converges even in nonlinear problems; huwever, it
does require two solut-oni for each time-step.
and
(15)
"
G(xy;t).
' n + at Fn
and
(16)
-=*
+ At G,
n+l
n
where #* denotes # on the free surface, the
subscript n refers to the time level, and 4t
is the time increment.
The values at the new
time-step are obtained from the corrector @tep
by
I
n+Il
in + "(F
+ FV
n
n
IV.
end
(i7)
9*
n+1
9* + '~G
+ C
n+l
).
Itncomputing F and G, the spatial derivetives can be computed from the finite element
30
n+l
In
6t(t)n+(At / 2)(tt
(18)
#yt'*xits-tx t-o.t,
Titt
and
on y -1
1).t
2
tnxt
(22)
(u)t-nx/Fr - pFr24xx
(#* )44t(f* ) I .
+ ad (4*),,+(af/2)
y(
yyU
ytfn
-#
-#
(19)
on y - 'n
(23)
on y - 7i
(24)
on y -7
(25)
4y
" 44yy~yy-4
4"denotes 0 at ymi
(
n+
tozt
at - t
and
F2
t
)t-11z/Fr2.U
- yy-44#z
t "x(u)tt+z(
)ttw-1/Fr 2+(u )
+ w (wtPt/Fr2_
+
t
r
xt
4
-44
on yy yt z z.t
(26)
5.
Eq. (26).
V
Nuerical Soluitimn
u
l
Finite element Solutions
Presure Distributions Problem. The problem of pressure distribution Is shown schematically in Fig. 3. We consider the distribution
.L0
p
((x)
2
o2
0 A
L
(27)
(27)
elsewhere
where P
57
5t
-#slrx
y-xzx~
y,
-1las
55 5face
,n y -
lxa'1
(20)
ls~
on y - T
(21)
s00
The downstream boundary is expanded periodically to contain the entire region of disturbante within the computational domain. The
domain is initially divided into regular
triangular elements with Ax - Ay - 0.05.
Equations (4) are solved by the finite
element method with the finite-difference timeupdating scheme for the pressure distribution
of Eq. (27). The linearized boundary condition
is also used to show the differences between
the linear and nonlinesr cases. Tv- time-step
used is At - 0.002 for the nonlinear case and
At - 0.005 for the linear case. In order to
make comparisons with the results of Hauseling
and Van Eseltine [5] , the Fronde number is
For the hydrofoil problem, the implementation of the Kutta condition is not simple in
the finite element method, the first-order
approximation
4 - Constant for all e t
is used with e
The evolution of the wave height is displayed in Fig. 4 for the linear case and in Fig. 5
for the nonlinear case. In Fig. 6, the nonlinear result at t - 1.0 is compared with the
corresponding linear calculation as well as
other solutions. These results show that the
local steady a
e solution can be achieved by
the proposed
-dependent approach and
indicate that the numerical scheme developed
yields reasonable numerical solutions for the
free surface problem. The numerical test
given here is only for small heights, but the
numerical scheme proposed can be applied
easily to cames of larger heights by restructuring the finite element mesh system.
The mesh generating scheme we have developed is used to obtain the initial finite
element mesh system. As time increases, the
boundary elements near the free surface follow
the free surface geometry and the mesh system
is rearranged when the surface elevation becomes large. The initial mesh systems are
shown in Fig. 10 for the ellipti: .ylinder and
in Fig. 11 for the hydrofoil.
A-
(28)
between such nodes was Ay - (A + 1). Successive over-relaxation (SOR) was used to solve
Laplace equation in its finite diffarenre
f
toni
form.
(3
+
n-I
n
+
- 4
-2
e)/2&t
(29)
310
-..
which include the trailing edge. This approximation works well when the element si7es near
the trailing edge are relatively small.
used to seek steady state -olutions. Otherwise, phenomena such as the wavebreaking as in
this ease may occur. Another important indication is that numerical solutions fr threedimensional problems other than very simple
ones tie feasible with current computer
technology.
pT11dx
Frf
Fr
VI,
been demonstrated by numerical solutions obtamned for a two-dimensional pressure distribution problem and a three-dimensional
accelerating strut problem.
a. - c cosh 6
and
where
bm
2 - t sinh 2
c - a2 _ b
. -
and
Conclusions
r + ln[ (a+) /c .
h
ial
element
inner
The
The inner
has triangles as basic elements.
outer
boudary
that
aetchregion has a square
em the inner boundary of the outer region.
dimen
uiial
flow past an elliptic cylinder is
Su.:h a scheme can be applied to problems such
therefore used as a condition at this boundary.
the
The application of any time-dependent
The initial conditions for the problem
The initial mccelarat ionThaplctoofnyim-endt to find the steady state solution of
at rest.
conditionscn aapproach
n
used is g/lO. The grid used to solve this
problem was uniform with spacing A - n/50 in
all three directions of the elliptic cylindric-
l coordinate system.
lea,
the spacing
and those at y
problems.
Re,
1.
2.
311
~-.
.
-..
~.----,--.-.-.1"'
'Numeri-
the
C.,
Salvsen, N.
nd von Kerek,
"Comparison of Numerical ark Perturbation
Solutions of Tso-limensional NonliceAr
Water-Wave Problems," J. Ship Rea..
vol. 20 (1976), pp. 160-170.
-.-
3.
4.
5.
R. T.,
Haussling, H. J. and Van sseltine,
"inite bifference Methods for Transient
Surfaces,"
Free
Potential Flows with
First nternational Conferenceon uaerical Ship Uyrod'...ics, Galthersburg.
rylad, Oct. 197j.
6.
17.
20.
21.
22.
23.
24.
p. 234.
I0. Nichols, B. D. and lirt,C. W., "Methods
for Calculating Multi-Dimensional,
Transient Fros Surface Flow Past Bodies,"
tiloallConference on rHumeri"M
Gaithersburg,
al Shin vdrovndamicl.
Maryland, Oct. 1975.
11. Chan, R. K.-C. and Szreet, 1. L.. "A
Computer Study of Finite-Amnlitude Water
Waves," Journal of Computtional PhySics,
Vol. 6, No. 1. Aug. 1970, p. 68.
12. Chan. R. K.-C., Street, R. L.. and Prom,
J. I., "The Digital Simulation of Water
Waves: An Evaluation of SUI4AL," acturp
Vol. 8 (edited by
Notes in hysic,
Ehlers. J. at at.), Sprinper-Verlag,
Berlin, 1970, p. 429.
Street, R. L., Chan, R. K.-C., and Frc'n,
J. E., "Two M&hods for the Computation of
the Notion of Long Water Wavest A Review
end Applications." Proceedings 0 the
hedsCz
Italy, Aug. 1970. v. 147.
e., Rom
14.
13.
Chan, R. K.-C.,
311
25.
Free Surface
~Element
26.
27.
28.
4.d.
31.
i
.A
xml
fRsrcuigEeet
near the Free Surface.
(
y"7(,;)
30.
, 7 F(.,yi
"'(-Y
l0~
.I
NumberLierFeSufcCodtosFr(r1.,0
Elemrent
,7
~~~~~~~NodeNumber
LferFe
ufc
ol~,OF'4f
.1
oOO
1.0.0
(0)
(b)
t. 130
(d)
(a) Six'triangular elements, one interfor nods, (b) Ten
triangular elements, two interior nodes, (c)
Six triangular elements, seven interior nodes,
(4) four quadrilateral elements, one interior
nods
Fig. 1.
0.02
t'2.0O__
0
-002
Height Computed
Fig. 4. Tim Evolution of Waive
with Linear Free Surface Conditione.
Fr (4")" and a 0.01.
313
- 0.0)---Re
0..2
soe
Rateof W(rk
t:0.
Growoth
Energy
t'0.5
1.0
0.02
Frand
Fig. 5.
Fny
u~t,0.~~0.02u
.01.U'1
X_-05
\.
X=0.5
-001~
057-
.01
and Fir-(4r)_11
V2,, -
a Ionlnasr
aolution at L - 1.0.
b
L.
314
Iuit
-~I,
}~
FigIl
315
EllipticCylinder
t-.5
0.0010
t-10
o.O
\ oo000000
IL- PresentStudy \
0.001 0 Hojnslitnq8, 8riEseltln
1.6
1.2
Oh
0.4
* 1~
Time,t
t.0.-02
ig.
-05.5
..
~.-]O02
Numerical Solution.
Fr
.0
-(41r)k
0.01.
16
Hydrofoil
0.001
I' 1.
~~1
~
__________________
_0.8
Time,i
(4oY)k and a
Fr -
0.01.
-0.00
0.1
-0.5
0,4
--
002
Finite Differemc
Finite Elemnent
000 -
001
10
0.0
20_
.0----
0.02.
Fiit Elementtn.t5dtat
00
001
---
-001Psiio
,.
0.2
LQ
Longi
t t-0
Wveproils
10 ad
Fig. 17.
.0
att
10
-10
318
Comparison ci frt~t
(r-k
Fr-(v)an
Element and
.1
-0..
Distance, x
0.6
-0.1
-- 0.0012
Fig. 18.
-0.4
-0.5
17
-0.0020
-- 0.0008/
--- 0.0004/
-/1
*--C
Contour Plots of the Free Surface Positiou nfir the Upstr.m End of the Strut at t
-0.05.
Distance, x
0.4
0.5
0.6
-0.10008
-- 0.0016
C
.4in
Fit
19
Contour Plots of the ?re Surface Position near the Dovuetream gad of the Strut at t -0.05.
317
0.01
.=0.05
0-
-0.01
.0
t =0.0555
S0-
-0.10
_--___
-----
I-
i =0.05878
0.05
0.04
003
0.02
0.01
318
Abstract
The control volume approach is used to ohtain a finite-difference scheme for a Lagrangian forazlatio. of inviscid incompressible
flow using an irregular triangular mesh. The
method permits grid reconnections and allows
local vertex addition and deletion. Algorithms
are presented which onserve divergence, vorticity, mass, momentum and energy even during
grid restructuring. Examples are taken from
simulations of shear flow and flow overshydrofoil in which the restructuring algorithms are
crucial. Although the structure of the code is
highly scalar, techniques are outlined for producing efficient code even for the new vector
computers.
together with the conservation of mass, rmomentum and energy. There are, of course, many
possible ways to design finite-difference
schemes for these equations. However, it is in
general not poscible to determine which ap5
proach will be successful.
For the case of
triangular grids, and in particular reconnectlog grids, there does not exist a literature of
proven techniques. Therefore the method chosen
for SPLISH was the control-volume approach, in
which the finite-difference equations are forislated to satisfy the conservation laws macroscopically, over a computational cell. In this
way the conservation of physical quantities is
explicitly satisfied by the scheme at the outset, and rcrections for non-conservation are
eliminated. .
I. Introduction
The hydrodynamics code SPLISH is designed
for Lagrangian simulations of transient freesurface phenomena. The present version of the
code was developed for inviscid, incompressible
flows in two dimensions. The method uses a
triangular finite-difference grid in which triangle sides are aligned along free-surfeces,
interfaces, boundaries and the perimeters of
submerged bodies. The grid internal to these
surfaces is left free to reconnect, adjusting
1'2
to the time-dependent flow.
In addition,
vertices can be added or subtracted as they
accusalate or become sparse in convergent and
divergent regions of flow. The added flexibility gained through such grid restructuring
permits the application of Lagrangian techniques to large classes of problem which were
formerly considered solveable only with the aid
4
of diffusive Eulerian rezone methods.?'
For
example, tha simulation of shear flows about
obstacles are possible with only local changes
in the grid. This paper will present the formulation and the motivation of several such
grid restructuring techniques, the algorithm
used in implementing them end examples of thai:
use in SPLISH. Because the lack of global ordering in a reconnecting grid is a drawback to
its implementation, a discussion of techniques
to produce more efficient codes is included.
Examples will be given of calculations performed on NRL's TI ASC pipolins computer.
tv
This research was supported by the Offi:e of
Naval Research.
31e
dV
o-
dt
.
iffL
(4)
V2
~itot
Figure 1. Definition
F~gue 1 of
ffnlton
a control volume
olue about
aout
an interior vertex, Vl. The area of triangle J
is apportioned equally to the control volumes
and V .
abut V
2 aintegral
V
That is, the pressures at the vertices are
iterated until the resultant triangle velocities reflect a divergence-free condition for
each control volume. An obvious construction
for a control volume for this application in
shown in Figure 1. %be vertex-centered control
volume is defined by lines extending from the
triangle centroids to the triangle side midpoints. This permits a unique, uniform and
complete tessalation of the entire computation&1 region. The control volume for each vertex
contains exactly one-third of the area of each
of the adjacent triangles. Because pressures
are defined as boundary conditions, the control
volumes are altered near boundaries as shown in
Figure 2. In this way the pressures at vartices near the boundaries enforce a divergencefree condition over the additional area as well.
()
V,
V2
V5
(b)
V,
V2
rigure . Conservation of vorticity while advancing vertex positions. The triangle veIncity is altebrd such that the V'dL line Intsret ce triution within the rotated eand
stretched tringlA remin the s, for each
vertex.
320
Thus 'ar we have shown only that a logical extension if the control volume approach is
applicable to a eneral triangular mesh. It
formulations
can lead to finite-difference
U
which macroscopica ly conserve approprinte physical quantities, rgardless of how irregular
The real utility of this
the mesh becomes
approach can be seer, however, wen we allow
the mesh the freedom to reconnect,
111. Reconnectios13
,3
..............
U
-.-
Despite the simplicity and physical motivation of such an algorithm, it is nor obvious
that it is the preferred one. The expression
for a general
1 TKI ij
;x(rc-ri)
r4
+ 9i+l ix(rirc)
x _
321
!p
4b
yc ix(r,+l-r
cv(
Vlcv-
t o )
1+
I-,
(b)
j+ 1
3
Figure 6. The reconnection Algorithm to preserve diagonal dominance of the Poisson Equation. The triangle and %ertex labelling user
in the Polsson Equation is show
in el.
Fig+
ore (b) iodicates the angles 0 and 0 used in
the reconnection test for the line from vrrtex
c to vertex l.
a
and is always negative.
A i+J
(6
reconnection. For exampls, to keep the vorticity and divergence conserved the portions t
the integrals
. AI
14*es
i-I Yc
Vx
- dA
Figure '.
The alteration of control volumes
by a reconnection. Portions of the control
volumes about vertex I and vertex 2 are shown
before and after the ieconnection,
jcot 0+ cot o+
;(b).
.22
VT
where
V
V
V V
Rwhose
BY
VT0
"
FX
VLxl
(V
Ly/
R2D
A,
R-C
2A\
2A
R-C
-A
R'D
RA
2A
2A
R.A
RB
2A B
Balternative
2
R'B)
(10)
D
L
R
AB
Figure 8.
Te
*,
..
+RAL
- Ap +-
"VB (I
whore
A-
AN + A
A + A1.
Figure
SPLISH.
.in
(Q)
mh was
.2
MARKERS
5.?IS________________
LIS
I
SP
__________________
The fluid flew near a soeratrix is another area in wh~ch traditional numerical La5rantgimn treatments fail. Figure 11 illustrata@ a sample grid for a aubstergod hydrofoil near a free surface. The flow it,directei
to the right Initially with velocity U. Clearly of the flow develops vertices will tend to
accumulate at the i.rward stagnation point. At
the so time the vertices on the hydrofoil
will move withs
the flow alonit the hydrofoil
and accuausate (by pairs) in its wake. After a
very sbort tim the griddtng wilt deteriorate
I
2
This claim requires elaboration. The remoal of a vertex implies the alteration of
four vertex control volumes, one of which is
removed, and such a drastic change does not
seem consistent with a mere change in resolution. Figure 12 illustrates the triangles before and after vertex removal. Before the verthe
tex is deleted the relevant contribution to
vorticity integrals about each vertex are
3
_
)
"
+ Vk
2
(r.r 4 )
2
C
+
+V
(r,
V
V di
4
g3
(j
-r1 )
2r
2
+
2
.dl-
(.r)
+ -(15
2
-
_ _
d'-"V
r)
2
V..i-
A4 V4 - A1 V1
)
_
_ _2v.
"
and
+(14)+1a
Eliminating I' t2 and
(16),we have
where triangle 4 encompasses the three triangles 1, 2, and 3. Since the mass of the resultant triangle can be defined in a similar
manner, the momentum of the larger triangle
has not bean altered. Therefore for both addition and subtraction of vertices, the larger
triangle aets as a control volume in exactly
the saw sense as the quadrilateral for the
recormections. What has been lost is the information about the behavior of the pressure
gradients and ,elocity gradients within the
triangle. This information has been averaged
out and replaced by a linear vartatitn across
the triangle. All we have suffered is a loss
in resolution, exactly what wu $et out to do in
removing the vertex.
i i.r) +)
+
vk"
'
+k
"(T4
4r
) V"
'V"4
and
(r.-r2 )
(=YI-Y)
1-2,
2
12
(17)
-V + A
Aj,
Sbstitutin
+
i
algebra,
V
k into
32
sens
directions, and resolution of the leading portion of the hydrofoil is being lost. A point
may be added along the body as in 13(b). The
result may be viewed as the addition of a point
within a triangle, but one in which only two of
the three sweller triangles survive.
A JA,
Ak94/A,
A i,~i/ALl
where
- F
(b)
since A + A + Ak*A
tc)
For the case of an added vertex, the vertex control volume integrals are trivially left
unchanged, and the added vertex initially carries no iorticity. Vorticity can accumulate
about the added vertex only through reconnections with triangles having dissimilar o. Thet
is, vorticity is generated only by density
gradients, as expected.
Therefore, vertices can be added and subtracted rithintriangles while conserving flow
properties exactly. In both cases the usefulness of this result derives from the reconnection algorithm. Used in tandem with addition
and deletion within triangles it provides a
general algorithm for altering the grid without
disturbing the fluid flow. It has already been
shown that reconnection used after addition t
a vertex at the triangle centroid liberato .
vertex in . conservative manner and permits it
to behave no longer as the centroid of the triangle. The process can also be reversed. The
- isolate
reconnection algorithu can be use'
any vertex within a larger triangle. Once this
is accomplished the vertex can be averaged nut.
(5~ ((b
idrel(a) illustrates the reverse aitustion at the rear of the hydrofoil. Thernew line is dran to enclose the unw~antedvetLex in a triangle. In Figure lh(b) the vertex
'i_
ved from the body, leaving the way
clear to add a verL- in the fluid, if needed
to preserve resolution, , in Figure 14(c).
the use of the control volume approach has
nade possible the dynmic aittion
there,
and subtraction of vertices exactly wi.,. desired and in a fashion which locally and globally conserves the properties of the fluid
flow. The combined use of local resolution
alteration and reconnectiun algorithms permits
Lsgrangian calculations of extremely compli-
~V.
Efficiency
Figure 13. The addition of a vertex at a boundary, The vertices on the bounday are moving
along opposite sides of a submerged body (a),
and resolution Is lost for the leading edge of
the body. In (b) a new vertex is added on k,,
boundary. The ol4 triangle is delated and
new triangles are added.
328
members Instead of performing r?n whole sequence of operations in turn for each member of
an array. The ordering of the vertices becomes all important, and a highly disordered
code such as SPIASH is almost totally unsuitable for efficient operation. However, despite such obvious problems, the entire SPLISH
code has been optimized for the NRL TI ASC
vector computer.
Although such increases in speed are encouraging, they are not the final solution.
Large calculations will require faster solvers.
The most promising approach is through the use
of direct solvers, rather than iterarive ones.
Although the matrix representing Eq. (5) does
not exhibit the ordering of rectangular meshes,
it is nontheless sparse. Furthermore, if the
vertices are preord2red by position, the nonzero members will lie along rather diffuse
bands. Recently, there hss been an increase in
interest in fast solvers for such banded wqtrices and several techniques look particularly
The outlook is very good.
encouraging,"'
Not only is a large class of problems now
amenable to Lagrangian calculations, jut also
at a computational cost per zone competitive
with other techniques.
However, even iii this situation some increase in speed may be gained through efficienc coding. Clearly s good deal of the time
in the reconnection algorithm is spent in testing each line for a possible reconnection. In
general, very few of the Iines reconnect for a
given timestep. The flow is following local
streamlines. Therefore the test can be vectorized provided its output is a list o' lines
which may want to be reconnected. Each of
these ffew) lines is then passed through the
scalar reconnect routines, in which they are
retested and the reconnection performed if it
is still desirable. An iteration -rough
this procedure may b desired, but in most
cases is not necessary since most reconnections
occur remote from each other. In no realistic
case tested were more than three iterations
required to reach the final grid.
References
1.
2.
5.
4.
5.
6.
7.
327
7i
.,'.'|
'1I
Feb. 1977.
9.
DISCUSSION$
of two papers
Invited Discussion
direction is nonstable.
Aad J. Hermans
Technische Hogeschool Delft
The authors should be congratulated for
their contribution of numerical tools to be used
inship hydrodynamics. Both papers present
methods to be used to solve the complete nonlinear free-surface problem in two-dimensional
problems, while the paper of Yen et al.presents
some results inthree dimensions as well. Up
to now these problems are considered as too comlicated for solving with purely analytical tools.
Papers of for instance Ogilvie, Dagan and myself
show that the linearized Kelvin-Neuman problem
leads to erroneous solutions at low Froude number, at least in certain two-dimensional problems.
For three-dimensional problems the same errors
are expected locally. Therefore it issuggested
that one take into account the non-linearity in
the free-surface condition,
The paper of Yen et al.brings up some questions as well. Inour paper earlier this mornIngwe presented a numerical technique that may
be modified to solve the same class of problems
as treated here, but in stationary situation.
In shallow-water problems it turns out that the
mean surface elevation infront of the disturbance is different from the mean elevation
behind the disturbance. This difference in mean
level in the steady-state problem isdue to a
phenomenon described by Brooke Benjamin and
drawn to our attention by a paper presented at
the IUTAM meeting in Delft by Salvesen and von
Kerczek in 1975.It turns out that ifone starts
at rest and gives the disturbance a constant
velocity suddenly, hydraulic jumps move forward
and backwards with different heights. These
jumps have to be found if one describes the
initial-value problem properly in the snallowwater case. Especially in the supercritical
case this effect is important. Inthe deep-water
case this effect isof no importance; therefore
the results look reasonably good. However, I
doubt whether inthe shallow-water case the
correct physical solution will come out of the
numerical treatment. At last I would like to
remark that the authors in their paper do not
pay much attention to the Kutta condition. It
may be of interest to present more details
about the vortex that is left behind and Its
influence on the free-surface elevation.
Discussion
b-ywa--ngune Bat
of paper by S.M. Yen, K.D. Lee and T.J. Akat
330
Ck
Author's Reply
Abstract
This paper describes some new and
accurate methods for the calculation of
the form- of steep gravity waves, and of
the time-history of unsteady, breaking
waves,
For steady waves, one method has
been to use the small-amplitude expansion due to Stokes, but introducing a
new expansion parameter which (unlike
the first Fourier coefficient)
increases monotonically over the permWith the aid of Pada
issible range.
approximants, satisfactory convergence
can be obtained with the use of about
10 terms.
These calculations have
been facilitated by the recent discovery of u new set of quadratic relations
between the coefficients in Stokes's
opansion.
The author and M.J.H. Fox have
developed a different approach for
waves of nearly limiting height, where
the wave crest is still rounded. They
have shown that the flow near the crest
tende to a certain asymptotic form,
is proportional
whose length-scale t
to the (non-zero) radius of curvature
at the crest. This asymptotic form is
the same for steady waves of any type,
whether in deep or in shallow water,
Using this as an "iruier solution", and
matching it to an "outer solution" representing the rest of the wave (wavelength L ), they have shown how to describs a steep wave by an expression
involving two terma onll for sufficiently small values of XL
. The
expression derived for the phase
velocity confirms very accurately the
previous calculations by Padi sums, and
shows in particular that the phasespeed in not a monotonic function of
the wave heightt the highest wave is not
the fastest,
For unsteady breakers, a new and
accurate method was proposed by the
author and developed in collaboration
with E.D. Cokelet. This sei-Lagrangtan method uses the values of the coorand of the velocitydinates X,
I.
Introd~gtAo
332
-how the particle velocity in the forwards jet may exoed 12 times the
phase-velocity for low waves of the
same length ().
1.0
Here we shall review recent calculations under three heads (a) steep
symmetric waves (b) the deformation of
the wave which lead
to breaking and
(c) the flow in whitecaps after
breaking. We also describe a new calculainstabilities
normal-mode
tion of the
of steep gravity waves, which indicates
that there are two different types.
First, there are subharmonic instabilities of the Benjamin-Feir type. These
tend to modulate the wave envelope, so
that the difference between high and
low waves constantly increases. Secondly there is a local type of instability,
concentrated near the wave crest, which
leads directly to plunging. The rates
of growth of these instabilities have
bee.1 accurately checked by the independent time-stepping metlhod described
under (b).
.....
'
1-0
>
I
I
1.090
042
0-44
0.43
WAVP AMITU09
0443
t)
II.
------
and if we write
Schwartz (1) overcame this obstacle
by -,in as expansion parameter not Cj
but o.C, each value of which defines a
unique wave.
+/
I/C
-H5 /c
,-
Soon afterwards (.) it was estabfished that not only each coefficient
4Lt,but also the phase-speed C attains
a maximum for waves lees than the highet,
in fact when oL.-O,43 6 (see Figuro i). The physical reason is connected with the fact that the highest waves
surface
are so sharp-crested that their
l-+,,,
.
then we have
+
-2
a.
333
kCo+
\ C, +"
I%0,
IO3
- ,
OL
A
. a.,
y I
+ I
-.
'"
Figure 2
from Longuet-Higgins and Fox (21.
of a steep, progressive gravity wave.
-II,
Asymptotic methods
For steep waves, the above methods
requiri summation of series to high
order N
where N may exceed 80. A
different approach (2, 10) reduces the
expreesion of c and other integral
quantities to only two terms.
Figure 3.
culeting
0 44343 - 0.T9
o
3
eWa (2. I't
+ Q0
30-370
(11)
(as*
(2)).
.I3
C
--
33
IV
C (t
(x t(X ))f
then have to solve, in effect, the wellknown Dirichlet problem, namely to find
on
tatona
prioican n sac (se
' I/,,Y
0
(.
everywhere inside C
~-plane
where
/')it denotes differentiation
following the motion. The last equation follows from the time-dependent
Bernoulli equation and the fact that
(VN
Hence
Figure 5.
The surface
In
the trans-
formed plane.
C
/
--
/07
//from
Figure 4.
plane.
-____
the (:.j)
integral we take
where in the right-hn
Since 0 Is known
the principle value.
everywhere op C ' . the right-hand is
given, and the equation is then a linear
equation for "'L./jn with given kernal
of initial conditions.
examples.
We quote three
It will be noticed that the computation points, which are also marked
particles, have a welcome tendency to
collect near points of marimum curvature, where they are most needed for
computational accuracy.
The method is evidently quite flexible and can be applied with a variety
C)
Id)
(j)).
Jfb
335
Overturniz
of a free wave,(d) to (g),
by a smoothly applied pressure
E/E.s = 1-67
0.61.
0.2
--
-0.2
-0.6
21n
Figure 7.
0.2:
613
"~
EIE .. n 613-
5W3
0.'
O.OS
313213
0. 1
0.
Figure 8.
(from Cokolet (,4))
of ti..
a,
2.
-4.
337
1.
V.
The instabilities
of steep
gravity waves.
nI+
-"
where 1,
4- are the velocity potential
and the streamfunction taken as independent variables; A , '(
represent the
unperturbed, finite-amplitude wave calculated
by Stokes's method
NORMAL MODE
or otherwise,
and
, V? are small, time-dependent
perturbaeions. The free surface is
given by V?
'
g,
where F
PERTURBATIONS
OF DEEP-WATER WAVES
_
-3
also is
--
--,-.-
.__--_____-2
form5
ZW 3
00
01
02
WAVE AMPLITUDE
A
In (6) and (2) the normal mod,,
were computed by resolving X, 'f
.,
and
each into a Fourier series in
.
The calculations showed that for suffC inetly small values of the ateepness C.A'
all normal-mode
of the unperturbed
'',ral,
and resemble
perturbations are
travelling waves v-,n frequency r * In I
where n is the wavenumbar of the perturbatlon. Thus ^w I corresponds to a
perturbation that shifts the unperturbed
we%. through a cona .Lai
phase.
Since
the phase-speed of the perturbed wave
L330
04 0-4434
(ak)
PE.
.. f
-toE
0.
ee
04 "4
04
5
o'
WAVE $7I0"ePIIs (.6)WAESIH14(k
Figure -)
(from Longuet-Higgina.(8)
The frequencies of normal modes of
oscillation (subbarmonics) having a
horisontal periodicity of M wavelengths, when nNv $ . 7Te real part
of the frequency is shown as a function of the steepness ok of the
unperturbed wave,
40
4t
04
.3
Figure 11
(from Longuet-Higgins, (~)
The ratej of growth of normal modes of
oscillati on (subbarmonics) having a
horizontal periodicity of f" wave. The imaginary
lengths, when m~part of the frequno.y is shown as a
function of the steepness (J?~ of the
unperturbed wave.
': j, 2/2)
Time-stepped calculation of the growing perturbation
Figure 12.
is the calculated profile, as it
.
On the left
when oJR .Q.3
develops through one half-cycle or the perturbation (time increases
On the right is shown the perturbation magnified
downwards).
vertically times 20.
340
3-1
Figure 13.
(continuation of Figure 12).
Further development
of the growing perturbation (*I/;Z 3/2])' when t.i.
.
(a
tions at tifses L- 0 and 'A/ a,
Then the
half-period) respectively.
apparent rate of growth j'was taken
asn
where
Jdenotes
.1
tis
I~
-02.O.2
wca 1.51 giving
lde
of It/
(see Table 1 ) compared to the *alue
obtained by the normal-Op0; At
mode analysis.
0.2
between 9.x
Q,
fficient C~
w'an also calcula ted, and
and JN4 -tAi
The max-.
maximised with reopect to A, .
mass correlation coefficient
1)'.
(see
Tabld
case
0'039
was in this
2.7
a.,
2as
3.0
3.1
2.t
3.3
wcs
Theycorrelation. coefficient C(
only slightly lees than for the 1 creasing soda,
Figure 14 shows a close-up of the
wave profile of Figure 13 during the final stag3 of overturning, as seen in a
frame of reference moving with the speedC.
T~able 1.
Normnal-mode calculation
0'10
0.0
0'0
0.25
0'2
0.8
of growth of perturbationa.
1/2
3/2
'2072
'2153
15.17
11.41
.0000
.0000
90
90
1/2
'2098
.2406
14'57
J3.o6
'0003
'0000
90
90
(1/2, 1/2)*
(1/2, 3/2)_
'2150
'2150
14.61
A4.61
'0132
-'0132
90
90
(1/2, 3/2)+
(1/2, 3/2)
-1931
.1931
16.27
16.27
*0234
90
-0234
1/2
'1804
'1309
37.41
24.00
'0000
'0000
17.07
'0000
g0
o064
120
*.3/2
(3/2
0.40
1/2
'184o
o041
(1/2, 3/2)'
0000
*Ooci
-'0001
.996
*0001
'C1
-988
~T
'985
'0138
-'0,125
.800
-'0175
'993
'875
-'0009
-'0040
* 990
o99
.254
90
90
120
-999
- 996
1T
',T
0.5
*a
-85
o.o65
.96
-o65
the
reIua-mode
A cruci al tetuf
ot
othmoemode
analysi
wa crried
.S.
Accat the besi c amplitude o.01
ording to Figures 9 and 10, the mode at
this amplitude should have become neutrally stable, that is, the rate of
At c...t# 0.3?
cowth should be zero.
there are actually two branches, vhich
for convenience way be designated nV
Figure
~
also small.
oe)
tenurl
oeA
_
11a
_
Onth
ef t hetmestpedwveprfle
; __eopeto
n txrih
is
thetrainelre vrial yafcu 0
34
Figure 16.
"~~.)t 0a?- I.
*the right is
On the left is the time-stopped wave profile;z
the0 perturbation enlarged vertically by a factor 5.
344
References
()
()
()
5*4
5.t
0.9
1.0
1.1
1.2
Po
,_
o.1
1.3
Ro
. on.
. 340 (1974)
1;71-493.
.4
()
(6)
Longuet-HLiggins,
(1975)
Soo, Lod. A. 342
M.S.
The instab-
VI,
(1978).
(2)
(Q)
(2)
(1978).
(0) Longuet-Higgins,
(f)
345
(12)
Breaking
Longuet-Higgins. M.S.
waves.
Proc. D.C.A.M.M. Symposium, Lyngby,
Copenhagen, March 1973.
(1a)
346
Abstract
Y0
a
(
A3
F
1. Introduction
Nonlinear free surface problems are
of importance in many ocean engineering
contexts. An example is sloshing of
fluid in a ship tank. Other examples
are the influence of steep surface waves
on marine structures and the slow drift
oscillations of a moored structure or a
low-waterplane area large-volume structure in irregular waves. Linear theory
is commonly used in predicting waveinduced motions and loads on a ship. But
nonlinear effects cannot be ignored in
extreme weather conditions or for
extreme shipforms (for example ships
with large bowflare).
Nomenclature
(x,y)
O(x,y,t)
t
r(x,t)
g
to(t)
4UWlaM$5s5a}jI
In this paper the boundary-integralequation technique is applied to a nonlinear two dimensional free surface problem with a body oscillating harmonically
with forced vertical velocity in the free
surface. The amplitude of oscillation
is finite. We neglect viscosity and
assume incompressible fluid and irrotational flow. The exact nonlinear free
surface conditions are satisfied. The
problem is solved as an initial-value
problem. When the problem has been
solved for one time-instant, the free
surface conditions are used to find the
free surface position and the velocity
potential on the free surface for the
next time-instant. At each time instant
the velocity potential is represented by
a distribution of sources and dipoles
over the wetted body surface and the
free surface. Unknowns at each timestep
are the velocity potential on the wetted
body surface and thQ normal derivative
of the velocity potential on the free
surface. Those are found by solving an
integral equation. The numerical calculations are significantly reduced by
representing the flow far away from the
body by a dipole with singularity in the
centre of th body.
(i)
Oy-
g9 +
(y 2t
(
- 0
)+
on y =
(x,t)
(2)
- 0
on y - t(x,t)
(3)
(4)
(5)
Other possible.
initialvalue
conditions
would have
been
We write
the velocity
of
the cylinder as k (t)3, where I is the
unit vector along 0 the positive y-axis,
conditio onithe w-tted
teor
The boundary condition on the wetted
body surface can be written as
-n- 10 (t )
0tn3
M
(6)
2. Theoretical Formulation
3. Solution Procedure
Consider an infinitely long horizontel rigid cylinder of arbitrary crosssection that is forced vertically in a
free surface. Let the water be infinite
in e"tort and be of infinite depth.
Initially the water is calm. The origin
of the coordinat system is in the plane
of the undisturbed water surface. The
y-axis is positive upwards and goes
logVI- TTT-F-j-TY
we can write
:
Pn -
Pn)ds
n
34.
1:1
x-
7)
where
Further we can write
S'rS + S + SB + SF + S1
Here S is the wetted body surface, S.
=
vertical control surface at x s urface far down
vB horizontal control
in the fluid S there surface adw
ialc ylindrical surface of smalwith
Srurh
(xj,yl)
radius r I and with axis through
which is a point in the fluid domain.
The contribution from SB and S are both
zero.zr.Note that for the steady7 statefresfa.
case the contribution fren S. is not
zero,
We can now write
l'Yl
xy l!x
a{ (Xy)
=f
1 ) 2+
(13)
(-x 1 ,y)
SF+S
-log/(x-x
J(x 1,yl) =
(8)
The contribution from the free surface integral can be rewritten. For
Ixl>b(t) where b(t) is a large number
dependent on time, we can write
9
-FA
(x, y).
(9)
r+ y
FOORE I
OF D=
EXA.EOF SUOn
SRFAE ANO FREE SURAE
(.)
? )o .-.
{,x~)
2
f{@(x2 y)T._.j-y)lo_'(x.x 1 )
b
.J,!,Llog/x7x)
_
anbx,y)
'+(y-y l ) )dxl
-A
bdxEIxly
re
r spectively.
y=0
I)
(10)
In a similar wcontribution from
to -b in equati
J(xj,Yl)b1
I - 7 log/(x-x.)+y!1 dx
'-n(xj,yj)
x
(11)
I-
1n
-Xl+:
x1
-A(
'n
+A ,
+A
,12an
'+..1
1,9an (K99*
2____
+ X 1 7 +Y 1
-qnlyl)arctg(b-x
'j))
(x
Y(,y
22'
-
(12)
349
B1
' +
A22,1O(XY)
A22
22
22
n(V2 2 ,Y 2 2 )-S
22
(14)
Here
i=1,22
Aijj 1:8
2wK+
+(~ -)I d
I o"*l'R
Dny)
[IojjT,)r
a
-nxy
x1)j(j18
follows directly from the body boundary
condition.
']j_)
ds (x,y)
A
-j
A
A ijJ
,i,22)=
-9,22
Idsurface
ff
A.
Dt
-J [log/(x-i)
i+j.
where K=J when i=j ard K=O when
Furthet F is zero except when j=22. hen
(
'i+
F=- x 2:
2*2" ' ( tY
'i
)/A
=.
,i(6
Further
and
-(x
j Iln
j'
y j )C
DxF
jD--
22
+ E Ox ,y)C_.
j=9
(y
)iq((x-7Rl
V j- Ids(x'Y)
i*,22.
2rK -
./(-
" xnxy{ l
(17)
In the numerical calculation procedure the free surface particles will not
always correspond to midpoints on the
free surface elements. When the fluid
characteristic is known at the midpoints,
the fluid characteristic at the fluid
particles will be determined by interpolation, and vice verse when the fluid
characteristic of the fluid particles
are known. The time stepping procedure
is performed by the Runge-Kutta method.
It is possible that a less tile-consuming method could also have been used
with satisfactory results.
4. Force Calculation
(j-9:22)
CiJiji-l,22
l,8
f
A
(1-1,22)
where
C i1
(16)
ay
Dt
(15)
ay
+(y-7)'] ds(x,y)+F
.I
. -
missing in
his expromemop.
iI
can be written
F - -Ifpi'dS + ffpgyidS
(18)
2S
ThVO1l
2
p-P1 -pgyl=-2at -
Pg
(19)
-LtVdS - hIffVjV#pjdT
- fi
S+S+San
V
S+6+Sf
fIlVfI2dS
f
pfIf{V- + kVIV{'}dT
at
V
SF
SF
SF
+fpyd
+ffp'dS
t8 +kjV 11} dS
P iI_
S.s5t
n
S
.D
+pff(V
(20)
IVI'A)dS
ddfffV~dT
V
d
P =p;-- If OAdS+ If pgyndS
--S+SF
S+SF
- fffV.- T
-;J1
V44dS
S+SF
an
n + Oilt
IVJ';)dS
(24)
S
The second term in (24) was not included
in the formula derived by Salvesen (6).
(21)
d fffV~dT
"PE
V
-If
V.-tdS
S+SF
(22)
n-
fffVj0VtjdT
V
dd If OndS
(23)
-If Rd
S
+ ffpgyndS
S
We can write
-IfplndS
(26)
fffVjV.I'd1
(25)
dS+ff pgydq
S+SF
ffPgyndS
SF
a +SIV, 1)AdS
If
--S+SF
$1fpgyndS
SF
5. Linear Theury
In order to gain some confidence in
the nonlinear theoretical prediction, a
361
This implies
wave componentm.
outside
- 0 cn y - 0 the
body
=t.gfy
3dS
0
0
(27)
(
F3
eiWtf (t)dt
(36)
3
0
--iwpfO(x,y;w)n 3dS-pfO(x,y,0)n 3dS
S
S
(28)
0(
3(
33
33
(37)
-PfO(x'y,0)n3dS
(35)
accordinq as w > 0
2.
AL4,
0 ;
ar
F 3 (-w) -
(38)
_F31w
(29)
0
iw
Y0 (w)-ieWt90 (t)dt
0
where i is the complex unit.
(30)
a circular cross-section
The integration is only from 0 to -Consider
with axis in mean free surface. The
because the variables are zero for t
radiueisa. The forced velocity i
The inversion formula is of the form
wrtte
O(xy,t). -10
I
t(x,yiw)dw
(31)
(39)
O(t) - cost
io(t)
where
(32)
e-'tcoellt
(40)
We can write
(see (30))
30
-0 on y
n, on mean position
of the body
(33)1
(34)
-Pf#(x'yO)n3ds-A33
52
(41)
xP
(42)
-2si (residue at
Thf
.(-A
B 3 3 (w)W +A 3 3 ()
(0)-
33
1_-- }
P3 (w)- - w .
0 C
(43)
when w > 0.
1
7
0
f e 'F_
f (t-2Re(T C
3()dw
Mei
iwt
(4
(44)
z_.
anc
i
B3 3 (i
A3 3 (w)Lp 2 I
4g -2) when w
= -2(
31rWFa
(Q)
(45)
"
inuation of
Here A1 is the analytic c
A 334+-iB along the ray ary )= -arctg()
w 33
A can be calculated in th' way presentd by Maskell(10). A computer program
based on his procedure has beern ,repared. Numerical results for A are
shown in figure 3.
w)o
Im
(ww)
3
/
p+a
pfa
/
FIGURE 2
DEFORMATION OF THE CONI'OLR
OF INTEGRATION
'
Cauchy's
',
L20
3I
' -1
-mm mm m m =mm~m
__
--
FiGURE I
CALCULATION OF F'OC
theorem
...
C)EFFICIENT
_', 1-051)
A I (W 1q)
P,
UnKayiay 3 2 n2..the
- yf
(lK-w,
~-r 2
Here Ka
and y
n2+.
face.
}(48)
=0,57725385.
were 60 totall,.
b . 7,73 whet'
wi)
1 - lffa(ln
Ka It
When
Lie*tasotter
P9 2 0 A,.
A1
TT
N,~oir
05Transient
part of force
(Lineartransient theory)
FORCED
HEAVE
MOTIONA, sin fit
(9
P-
in
20
30.
&a
r.t
FIUR
HYDODUYNAMIC
FORCE
ONCRCUAR
CYLINDER
LItNEAR
RESTORNGFORCE)
(EXTWOED
inteqrationpath.
6. Nmercal
esutQ/
Results by the nonlinear and the
linear transient solution methods are
presented in figures 4 to 7. The
ngu
-to0
theory
Transient part of force
tLinrio trarlz.rit
ter)A,
-5Noniear
theory A.
t0eo1
HEAVEMOTION
FORCED
11"
FOURE
A
Tn'
HYDRODYNAMIC
FORCE
ON4
HDOYAIFOCONCIRCULAR CYLINDIER
(XLDDLINEAR
RESTORING
FORCE)
00ICLRCYLINDER
1EXCLLJOED
LINEARRESTORING
ill)
Aatf.o
theory
Lieartvaesmet
Nonlinarthory.i. 01
-05Wo~oer
FOCEO HEAVE
.~.
tO 20
10
MOTION A, srnrit
,The
nt
HYDRODYNAMIC
FORCE
ONfCOCjAAR
CYLINDER
IE5CUN3E
LWEAR
WF5TMM FOCE
the mean free surface. The forced heave
motion is A3 sinflt weethe nonlina
results are for A where oF.rutoor
fortee
0f1
elmet
surac
-~Afeqailength
weereached
344
peoriod.
numerical results by the nonlinear method will depeaml ox, the ririrser
and the length of the #un race srol-n
The length of the elements must r't
exam"~ a certain fraction of the wave
The number of surface eileront-i
lengths.
will be a function of *Ime, j. to be
more precise, b is a functi''n of timer.
imatance, the solution p'nocedr'
becomes invalid when a surface wave ham
b. In reality the solution pro-
extent by varying b.
langth of the iree surfacetelements are
kee equal toN a.
Result for
A
Q/
I.AU and 0.5 are presented in
figures 8 and 9. The amplitude ratio
a
is 0.1. Based on these results
FORCC
10T104:
HEVE
A.mmThe
A,)
1.0vaE~i
TQb
773
a.
1. and a--0.1, 0.2 and 0.25
*....,
3'f.I
Results for
are presented in figure 11.
b.3D
3 2
'
-A
5I
FOURE
aFORCED
-05
-IsEXCLUDJED
.0.5
b-3,24
o3
0.3,
*.figure 12, Number of free surface elements are 40. Their initial lengtha are
0.3a. To my knowledge there exist
neither analytical methods nor modeltest
31)
2,0
HEAVE
MTM: Asinnt
HYDRODYNAMIC
FORCEON
CCLAAR CYLINDER
LINEARRESTORINIG
FORCED
HEAVE
MOTIONA
sint
b-773
b-&49
REULTTS
F L
AORCES)
:.,9
5 A
bQ
FIGUR~E
2phase.
~M.
Lec
SELUECE
b
04
THERESUL.TS
It shows
a.~
1)325'a
when.
9tt
"'
FIOLPE
12
IYDRODYNAUIPOF ONCIC~IAR CYLINDER
E.KCt"DELWEAR
RESTORINO
FOt6S)
J..w,.
n If
ID
n~.05
0$
D~.'INAE
,D~i~MN1CI
OF b
04 b
't_
- fails foL gitven b.
Thr
ilaJ'eys .-, somseambiguity Ivolved in daemnn
0)ALR'fluonce
wsien
FORCED HEAVE
MOTION
A~sinnt
the "nIuti.
_____
d (A33
(tW
Ft- 3 3 ( )Y0
(50)
to
A =Damping
'
ao
Aglmatd o
lets noninear.nethn
65s .,
.e
fit
FIOUE 13
HYDRODYNAMIC FORCE ON CIRCULAR CYLINOER
IEXCLULED LINEARRESTORING
FORCES)
canr t be easily explained, but it
should be noted that the results for
smaller amplitude ratios have a similar
behaviour,
nonlinearity in the free surface condition is much more significant than the
nonlinearity in the body boundary condition.
b) The velocity potential is represented as a distribution of sources over
the body surface and the free surface.
In the exterior problem both sources and
dipoles were used. In principle both
approaches are right.
c) In the sloshing problem the free
surface position and the velocity potential on the free surface are obtained by
following "points" on the free surface
with constant horizontal coordinate. In
the exterior problem we are following
fluid particles.
d) In the sloshing problem it is
necessary to introduce an artificial
damping term of the Rayligh viscosity
type in the nonlinear free surface condition, i.e. we add a term 4 to the
left hand side of the dynamic free surface condition (2). The damping is
small, but essential because there is no
damping in a potential flow inside a
tank. By examining the analytical solution of the linear transient sloshing
problem, we will note that transient
terms will not die out if damping Is not
present.
In reality (i.e. modelteats) a
steady-state oscillation is clearly seen.
is therefore introduced and represents a substitute for viscous losses
primarily in the boundary laer close to
the tank wall.
e) In the sloshing problem it is
necessar to solve the problem over a
substantial number of oscillation periads to achieve steady-state oscillation.
In the exterior problem steady-state
oscillations will nearly be achieved
within the first oscillation period.
An example of the calculations is
bhown in figure 14 for a rectangular
tank that is forced to oscillate harmonically in sway. The tank breadth is
7, Interior Problem
The interior problem, i.e. the
sloshing problem, has also been studied.
No details will be given here since a
There are
In many cases
this is no
limitation.
AT x.-0,5
T. 2 %.5%
03J
02,
01
0
Q2(
-03'
GA1
F3URE 14
std
prbstnle
Engui, I
4e.r.'Og, -
n
ntion
no
7.
qutpeer
Th al.o
,'usria
An
*nonthea
ntns
e andh
Urse
il
hip Resefrh,
Tt'ui
tmarked
ti.
nmOdrStay
trahitiui
Ra
iiu
eet. S t
th74
os1
V,
. a felui "'Ia
Math.n 2, 1 'it
10.
io
of.
.
,h
74.irfc
iv
:Med
Apree
b-94u.
,Ii
Maskel 1,
CtUt
I.
1. Thne
fre Fli e.
v 6loti
1
1,
t
Ph.. 303-'ta
1
oi Uivrriyo
12,
fluid
pp
0-i',.
'ec'tdi
qua't i'.'68.
np iniptre
in.
ainesd"
ni
ev[ils
Ith
ppli,21esJ,!9')
sloshing protion Butrbemsn
are
ante-'
cait
when
rm
thefre
suraced
oma andakel
thle ysrfacsen
a
nomaleto thew interDse
sectiintetwe
t free urfe and
hec
bod surfacxenaet cbertoallyexs
.foranked
oscll
g MhiResearich,
'it
hi
"i
olna
netanr
ihF
nr'u
o be pub-
'
1ee
''
poluiondagrecessr
welth tmahtno-.
-sof
osilos.l
t
wate onteDnmicas
A,
niutleiand Stroctures in
,;-26it Co9 eg,
onon
toa
'
iea
-tho
Teation hethn
wi
thel ast~
a fncton*
i
o tudy. The foineeail nuef thca et ios
ise copare withak ine eare inure
terstedtin thc gross-motindertedaa
Thin ito conetro th ratheria fwreut
poisn idis
ofeagntude. Th
n0)nar
lneer souton
free sufae
s
tyalo
on
theaf buihaoion. th sth istantce foi
denc ufalee
sutta
to the
ubr
ofuse
dutheteo
o
.o'-'
i.sen,
F,
Ms.
,J.o
o.,Uieriyo
acetr
1ylinper
oln
Frakuid'scillation
Wayhinnd.C.
o
36
lTeoryo
afree
This work was sponsored by the Office of Maritime Technology of the U.S. Maritime Administration.
Abstract
boundary location.
A sabstential fraction of
accurately.
The ship
action;
field.
on the ship.
This
I.
Introduction
time on a CID
coastal waters has created a need for predictive methods to analyze the
employed.
Chapter
Ill presents
11.
mathematical Formulalion
and
(2.2).
ship motion.
0(.1
div V
The method involves solving thu ruler hydrodynamic equation coupled with the rigid b dy
equation of motion for the ship.
where
fluid
treated an a reactive
-V
at
3V
in response
to
to
V -
d.1
rhJs
3M*
dVsI1.
Solution Techniques
(2.3)
Pdn + megJ
dt =f
px (p
(2.4)
where
n
o
o - surface area in the direction of V
v
s
= moment of inertia about an axis per-cacltos
n~
dclrtto the
tepaeo
pendicular
plane of
i
calculations.
Saicell
II=angular velocity about the axis of In
n
The motion of the particles is subject to
the fact that the particles are constrained to
lie on the aurZace of a rigid body. If (x, y)
marks the position of a representative inertial
marker particle, then
x
-X
+V
new
old
Xs
t -il p 6t
e y
(2.5)
u(i-l,j)
'
u(ij)
p(i,j)
y
new
' YoId
ld
6t + f
6t
(2.6)
yj
j6y'
where
i6x
P
p
P y
V ,V
i
la
(i+l)6i
e kreferenced
mash cells.
Now the true boundary consists of
a number of segments whose union is the total
boundary and has te
intersection, each
associated with a mash cell.
The second step
3Mt
In the free surface boundary cells the pressure is calculated by a linear interpolation
from the pressure in the fluid cells immediately belm it.
p
iJ-l
6y
is the
y Conditions
u.
Sk sh
kh)
4k
W
coah (kh)
!
ask
.1
(I-)
a
p ahere
L the required tor specified) free
surface pressure (normally zero) and a - 6y/6,
where 6 is the distance from the free siirface
"opa
i,j
. kh
i
h (kehw
where
*k
-
U0
wave number
o
k
t
os (kx-t)
n
i
a
g
wave amplitude
gravitational constant
=wave frequency
-dpth
d
coordinate
h - depth of water
x - coordinate in direction of propagation
t- time
w )
e-(S/
((3.1)(A/w)
6p
n,
172.8 h' 7
r
Full Cells:
"
691 T '
where
(o/8t
*n
x(t)
cos (mat C t )
0
n
(3.2)
n-0
Boundary Cells,
-here
x
6p
S0
associated
the
midpoint.
381
3e
i
!
ui-li,j) -
(6t/!Sx)6p
vii,j-li * v(i,j-l) -
(dt/6x)lp
u(i-l,j
a mesh parameter,
as follo ,
This equation is
imi,,i
nii,ii
pii.3)
6Op
u(l,)) * itt/dx)ip
,viij
# it/6yi6p
3h
s.-
ulated
esplI
iji
ah
Ah
t
it
35
I a h
h)
a.2
itude is
I.,/
Mg'
center of mass
maximUm velocity
in
the
flow field.
Due to tl.
3h/in -
In order to deternie
Th
pi
by interpolation, it
O.
Reactive Boundaries
Therefore, before
attempting
to detzrmine
pi,
pre3sures are
them in a
cedure.
fashion dictated by the inertial characteristics of these boundaries. Let us assume that
by
cells.
(x,
arc zero at
of these
Vb
and
Explicit finite
time zero.
parttcles.
A
. i
x Ai. It is assured that
init-ii condition is that the fluid and the
dx:/it
the
se
rents,
step.
boundary.
At the cor,:lusion of
tional
Let
and
be
and
ti
Then, i
in
iqure
orga.
n is pre-
3-2.
Is governel
Considerations
It
ndition.
Tis restricto
to
RI-6t
I dt
mn
I')
where
.
A
1 1.
is the vector foce acting at
X,
the vector
(metacentee)
.the
I....
at
c - the, ma
from the center of
to the point
accl,rator
it is
pointi.tg away
Irtro flc d
-
x
p
Since
r:'tat ion
J.
ia,
nmerlical
,,
due to gravity
33
spec'
viscous diffusion
stability [.qut.rs
term
are presera.
au
ax
_u
:;ya
,,
.5.
?Ie Fiel 1 3
IV.
Results
ar
veloc1ui
four horizontal
Results are
of the model
rif
setup is
test
shvwn in
Figure 4-1.
a
nt
tr-e
ti
;-tt
Figure 4-1.
3-2.
Figure
5.
Instabilities
Flow Chart
may arise if
sharp curvatures
in
As a general
mesh size.
there are
the ship is
rule at least 3 to 5
las
in
the boundary.
In otherwords
floating bodies
6.
iis asreh
a&siuzS,
Et
e
accuracy.
The
This value
11mmentum conservation is
also an
ot the
Nh
and
With
improve
(xt,
reference
be initial
yo)
of point
find this
be the forces
betefrsinx
where
to
in
x, y
(n
the presence
superscript denoting
Figure
4-3,
(equilibriam)
let
position
on
the deck.
and error.
smpottant
Vx, F h
y
IFadN
mtaenter
body free-
The mooring
dynamics as follows.
degree of mass conservation is directly dependent on tow stringent the convargence criterion
cutting down
system is
regions must be
tailt
uwrca
6.khnone
IWten once nasser
ical stability
(volums)
third dimension
As a result, slab
in the
of
be the height o
be
rap"
equations
form
38
(X,y
Cc
'z
metacenter
__________________________
9Ift
T-5.0
30 ft 6X-20 ft, 6y-6 ft, 6t=0.2 secs.
770 ft
T=15.0
no
n
n
(x,
yp
be the new position of point
p due to a translation of (Ox, Sy) of the
metacenter and a rotation of dO about the
metacenter,
n
P
T-19.0
then
x +6x
p
yo+
p
now define
sin d6
6y-j;
DX
I _.ncosd8)
and
DY
as
!n
DX
x-
6x -7sin
dy
6 - 7(1 - cos d8)
p
T-84.0
dO
(4.1)
(DX,DY)
By hypothesis,
a factor of 8)
6x-
Fr
,,
L~ldv
(xn
x)
6y_ (yn
Yc ,
(x , v
where
*[j
..
',
4kDX+
U
4kDY-
Kef f DX
iKaf f DY
re
Fiqure 4-3.
Schmatic of Mo-oring
Forces
Kef-
4_
the combined
eff
h
y
D
DY
K
+ Keff
y
F-F
DX
+ K
DY7sin dO
Ke ff
-K
.--
OysinO
(
-heoretLeai
tnepcted
s*
:AT. ft.
- 5.2
D.
n
if we use
Thus,
(Hd.I Tetj
n
M
F,
n
F,
n
Mn
in
computing the
c ,
0.0
0.4
Figure
IIo
0.I
4-5.
1.4
Smooring
Block coefficient
5.3
- 3.65
0.75
- 0.997
Mid-ship section coefficient
Stiffness of each mooring spring - 5 tons/m
0.
0.)
Figure 4-4.
0.9
-. ead/
).)
produce.
A still
lations.
which
65 ft.
water depth of
the
mesh.
Therefore, it ie necessary to generate a
slightly larger wave at the inflow boundary so
Computed
- - Model Test
m
mean - 74.0 ft.
1.2
2u
,
-I
'
In Reference [101, Verhagen gives a plausability argument that the standard deviation
of the sway amplitude is related to the
normalized eoctral density function by a
linear relationship as shown in Figure 4-8. He
conducted three model tests to confirm this
hypothesis as shown. We simulated this series
of model Lests and obtained the results shown
in Figure 4-8.
Nlp fe I-Ss_
Computfd
C-odliutand
est
- - Model
mean - 401.4
Sa
4.17
4
a,
Figure 4-6.
3.-n
FREE SURAC-AVE
Mean
a
COPuted
-._-Model Test
74.0 ft.
1.57 ft.
00
0>
/T,
ft' sec"
0i
sueary Comparison of
Figure 4-8.
O
I.
.Model
Concuilons
V.
o
VALUE
OK
'e'
Computed
predictive technique.
- - - Model Test
Ao FiniteDifference
njppenix
l.2
11a ft.
The finite difference notation usad in this
paper i
n
0'
"
49t.
un
representing
The difference approximation
for a
(2.1)
Eq.
quatiOn,
continuity
typical cell (i,j)
is
.
s
LEe
( n+l
(ul+
.'-
nj)
" "-.)
n+l
n~l
.
d.l"
"
, .i.
3r
Ul
+ gx
,-
PUY + VISX
FUY
+V
and
n
v
nil
vii
i
-iV-FV
[i
L
,1il,j
Ps
vsj
..
u~
+,-
"
ru~
6YgM i'j+l
where the convective and vi-ous fluxes are
defined as
rx
Ui+l,j)
(uil,j +
{z-v+
+i'T( i+ 1
L[(
+
iv
4dyL[(V"
+ a vi,
+
i
,Jli,J
)
i,j+/l
2.
+u
+
+
i+l,'-I
'i
-1
)Walls,"
i,)
j +ui,
)(vij
ii
46x
James A. Viecelli,
+4.
March 1966.
_ul
j +i,J+
+vi+,,)
+ 'a +
11
3. i.
R. Welch, at al., "A Computing Technique
for Solving Viscous, Incompressible, Transient Fluid Flow Problems Involving Free
Surfaces," Los Alamos Scimntifi.c Laboratory,
' u'J
" 4
X
'ilj
References
oUI"] U1.
-Y
i' , +u i J- I)
and
"
+ i-l,)
vli "
uij
i+1,j)
Patrick J. Roach,, C~ o
WA!j cs, Hermosa
)5.
Publiohers,
nsl&;Jm
Flui
Albuquc.-que,
Ui-.l,j
Ui-l,j+l
i'l,"
i,j)
46y
i.'
+7.
i
'
.......
..
.
..
..
...
...
.
G.
I. Bourianoff, "Numerical
Simulation of
Viscous Flows with
Reactive
Boundaries:
The Imp Method," Computationa
Methods in Nonlinear
Mechaiics, 1974,
Pp. 121-128.
Incompressible
10. J. H. G. Verhagen,
370
A
?
innivdeityoAisMe
Ar
Two-dimensi nal
past a semi-nfinite
slow
body with eonstant araft sH, is solved by
yepansion in a,fo l power
ries wth respect
bu parameter , related to he drft-baed
Fnrzl number F,-lc
.- This
re aeries
e
is
as-, vhere diverge,
but ca be suinuedby
st
p me.thods.
sard
Di,- ae
isrio
e
unique, .nd
one perofular , !!'iciest s-ation
procedure
yields R i-;contit ;ious : tw-e-free free
;:urfa"eIter.!tJ
-echniques are de-eloped
to procapp rs
!
tsolution
s a starting
point, to a
ng-ouso
lsunion posessint
non-lnrter wve:,
hiih cAn rwi-tefore serve
asg a stern f
ult!: n-: presented for
the steepness shie eav j
enerted
a and
fucton or F,
i
.s cosflt-uous wve-free
solution appears to
ist,
nd nuorigil
eviden0e p ut ebt th prtnce of a re-entrant
jet in the corres Ioie,
'hw-fw
t
robbl eb
p. ln far ecto
Shorizontal.
Sexpansion
owdHoM-
Aebstract
In the present 1,
Apprarpriateptrrtninrtorh
tniss solutioe'e
then used as a "seed" font a direet numerical
attack on the problim, using a form o{" 11owton-
v, we study further
san iteration.
This prucedure hs in prinhiple
quite independent of the lw-Froude number
asymptotic series,
ca
hd be viewed as ap
stndard nuerical solution of tie inw.redifferentn[ equation to vhi!h the complete
problem ta b reduced, at arbitrrry Froude
nresher
convergence of the thr tioriv Is very sensitive to the c.ii.e or
sterin solution,
d sucesa
has bcen tihied only by use of the suitmd series he such
a av
starting solution.
The Iterations apper
to ccnvergc extremely well for stern prow, but
not for bow flow.
n good
t hasaern
in practice
but values
atelgil
numerl
for an t able
he he
ser
These ters are quite uniqueLy determined. and
reande
ndon
of the sign oa
371j
Uairotica.iv
of orde
timesso&
steadily
ovng
..
st
i-in
wniaetd
.\
'
2. Mathematical Formulation
We assume a two-dimensional irrotationai
flow past a semi-infinite body, consisting of
a plane lower surface y* v -H, x* < 0, and a
-H + x* tan y which slopes at an
ftce y*
Figure 1 shows
angle y to the horizontal.
an idealized flow of this nature, which behaves
e
at infinity as a uniform stream U in the x
direction, witsL the free surface ultimately the
plane y* - O, x* , + . The fine detail of
the flow sketched in Figure 1 (e.g. the tagnation point of contact, and the absence of
waves) may not be representative of .he true
flow to be computed; indeed it is part of our
task to determine If these features are present.
Figure 1 shove a stern flow with U > 0; the
method is, however, equally applicable to bow
flows with U < 0.
.
YOu /s- P
Free Surface/
O.We
Ay"
.*-H,*- X
Y ,
4-1
C
X
Sketch of non-dimensional flow
Figure 2.
,-
'
(2.1)
(2.2)
and
- **IK
(2.5)
us
(2.6)
*
a(
) v y - 0.
=4+
14.8
a +.9)
* i,
of 7,
and solve for z as an analytic unc
in 0 C 0.
The free surface and the -iy are
= 0, on Which
portions of the streamline
(2.7) becomes
AVC
(2.7)
> 0. (2.10)
+ y') + C - 0
y(x
(cns "y,
O0 -1 <
0 0. (2.11)
cy(*,0;e).
C
Flat Bo
4ta
loping
ody
i,ree L'urface
_...L
*-
Figure 3.
Flow ii (,4)
plane
S-
H/L
y"-l,O;C) - C
of
LI
sF 1
number
Fm
g
= (-
N - y(-l,O;)/2c)
For
(2.23)
(2.23)
dp
(2.15)
(2.10) we obtain a non-linear singular integrodifferential equation for the unknow quantity
Y(Oo;C). , > 0.
(2.16)
- < 43
(2.16
Z'(f) . (frl)
'
0
, y > 11/3
1i'
-
(21M)
.
!2.13)
4,> 0,
--1, where
'
(2.17)
71
solution
0, the problem has the exact
n'(f)
(2.18)
=(lI',
irJ\pl
it.
x(f)
(2.2h)
We now assume
the Integro-differentia
equation obtained bythat
combining (2.10), (2.23),
12.9
Z'(f) - I ,
(2.19)
"
y(OO;c)
y (+)E'
since X
we obtain
X(f)(( w.
-f
0 as,
On letting
--
I
.-o -o
(
(3.?)
(2.20)
(3.1)
5-
(f}
(3.3)
\TY
(2.21)
.
O-AP
X(, - i0,-
*>0.
- JLlitl
(2.22)
O,
o
Thus
Y.* 0
and
x'(#)
ys
,., where
-S
3?3
J.m' n
,,.
2,3,1,...
,;,.
(3.4
(3.)
(3.5)
.
"uninteresting" in the sense that the departure of the free surface from its limiting plane
is required to be quite small, and the inevitable loss of accuracy of the asymptotic expansion takes place before interesting displacements of the free surface (including waves)
have a chance to occur.
sy
1f\(w)
.
'fLl
dp (3.6)
](3.7)
S=
obtaining
( )
)-
and
as
+(6
, "
a.
-s
Ti-")
,
_-,
J6
-'
-a
/
/
large.
(3.8)
__
(4.1)
r .
, --
0(k-l
O3
02
1, 135l
_..---02
(0
--0
----- * 6
05
0
0O
run
.a if duble precison
The progr
(.,9 figures) on a CDC 6400 computer, with
N'tOO, and wa
oun t to enable evaluatlon of
With .- figuir, auncy , laid
terati; up U,1 .
uptto yi 1
with h-fIgure accuracy.
4.
n It.1.~
I)
()
-
(3.9)
-02
-01
Figure 4.
Polar plot of
(*(.
dx
(4.2
~sbe
J1
x'
)dt
(11.3)
e(I)(n + a)i
A0E)-
lt(E
.9)
fucosrelin
'(t
ca
obtained by specifying different cuts it,
l
t
eutn
tecmlxtpae
solutions differ by a term which is exponentitly smell In the limit as e - 0 and all
have therefore the same asymptotic expansion.
The main difficulty about the converging factor
is therefore one of non-uniqueness.
Is particular we canereplace WN(t) in
(4.9) by either of the following two fuctiones:
W (t)
+
(4.4)
..It.> 0,
relation
"
Ft-
<i(-)
0',~
(4..10)
W,(IA -
or
W
W_(t)
N
t + 2lie(-sl'
'N11
1t , 0
(IA
zio
0.
N(.1
0.
ug e tate wit
valhs to
howeerthi
vale hs t be reaed ithcould
caution, as an accurate evaluation of the
constant "a'
based on only 15 term in dirficult.
We will therefore devellp the followlog arguments without aniy assumption on the
constant
a .
cowevrtis
(b.5)
z'i
0.
(E'
Z.
. (7
.a)I
V~
(46)real
The Infinite series
Fi r (n
a)I
ad
I
~;-r
~( J-~
s\
(4.7)
it-
ster.
lu-b)
e
u'
The function
a
and posesestrin
sr-ro
ie
coer
,%r
_ )(
c'x-C(
jr)
.
x)"e'T W
ee -e,
(5.2)
/
'i,.(*)I,
do
(4.12)
-1
E(
- E-I)
e 0 a-
(C.3)
1e4 Ei(
(5.3)
(k)
On the other hand el
is rea and cannot
therefore converge to a complex rumber when
Therefore the only possible
x is real.
choice in to cut the complex x-plan, along the
A similar argument hf.
real positive uxis.
expansion.
of the function
Useful experience can be gained by considering first the simple divergeot expansion
x* ni
log(l+z).
(5.1)
s~f
where a is a complex table.
When applied
to the series 15.1). the iterated Shanks
,
traLnaformsation
," conerges rapidly,
o longI
as 9
is not real and positive, to the function
Si'
surface, even
c .
A guni
the otsag.at on
4 do not give
I
where
35-
.0
2.0
T'~~~
FM0
Ja(4y~
3.0
0.
is a known function of
-0.1
"9
'g, e 0"65
FEquations
-0-2
4 b+ y(P)
lim
-0.5-
(6.4)
ye(b).
free-surface
Figure 5. Discontinuous
profiles obtained by Shanks
method.
y' = (y',y.,y.
2
(y,y
,y
y )
'..yN)
(6.6)
and
a
.x),
= (x,x ,x.
(6.5)
(6.T)
whose components are the values of the corresponding functions at the mesh points chosen.
The fist mesh point is at 0 - b and
i'.,where 0 , is a
the last one is at 0
large number.
The error inherent in approximating an infinite integral by a finite interval was found in most cases to be negligible,
at a distance less than a wavelength from the
last mesh point, for ., sufficiently large.
Using the trapezoidal rule and Monacella's
theorem to approximate the integrals, we get
relations of the form
N
F +
a
1,2.
+I
0
(6.8)
N
y.
,i
1,2.
N.
(6.i)
6. Stern Flow
",;
- O.
g(
(6.to)
b,
is expected to be
(6.1)
y;
y(O) - ye($),
0<
Wb,
where A
elements
(6.2)
- A'I(
is the matrix
of the matrix
\4/Jte
(6.3)
a)
Inversion
T
scheme was tound to be rapidly converaent, and a soluton of the algebraic equations
3"-*.
(6.11)
h
with (i,j)t
y' - 0
For the first approximation we take
Each Iteration requires the computatiln nd
Equation
known function.
y"(4)
in the form
be are-vritten
now is
where
(2.23) ema
-v)I
),
2e/2Z
...
0-09
0.07 0.06
005
"
W0-04
0-03
0"02
4 F5
y-90'.
6
F
4
.predominant
-----
____
0i
Figure 6.
10
iS
.J
constant less
20
Relationship between
iu tuid
Fn
Mid
376
20
t*0
Y/H
20
600
400
-10k
Figure 3,
7..Bow Flow
(*)
y*(#) . b 4 4 % #
F11 = 6.3
90,
,
This "solution" is clearly verj aporoxindeed, as the flow between the last
r i t shown and the stagnation point has beec
completely filtered out.
However, it is
fiet
37
.----.
--
,~Nou-tonian
-0.3
iterations.
Free ruace
-0.6
I-.
x~6
I~
____
it
--
tihetch of conjectured
jet-ilke bow flow
,with
that
y(x*'+ y)=0,
;
;7
(7.3)
p~o~J
Iir
A)
0.0
wtiSch
Figure 11.
Figs'r 10 .
Appendix
References
1. Abramowitz, M. & Stegun L.A. (eds.) 1964
Handbook of Mathematical Functions,
Appi. Math. Bar. No. 55, Nat. Bur.
Stand., Washington, D.C.
2.
3.
Baker, G.A. 1965 The theory and application of the Pad4 method, in Advances
in Theoretical Physics (K. Breuckner,
ed.) Vol.1, p.i. Academic Press.
Yn =
5.
6.
S3' 5.,+
'[Y..
Y,.
+ 0(x,., Y,..)
,
y
2'
+
+ yx!)x. +
(y
2x
+ y!QY.
+Y1%Y - +
(A.1)
The leading term in (A.1) is normally domin.nt for large n, and thus the growth of the
coefficients is asymptotically given by
On the
susceptibility of a ferromagnetic
Proc. R. Soc.
above the Curie point.
London. A2,O0,2114-228.
JMatb.i_
sequences,.
1-42.
y.
s
x',
s
n *
(A.2)
- R(4),
(A.3)
Lquation
for any arbitrary function R($).
(A.3) is a linear mixed boundary condition for
sO,
0
y(*, ) on
the potential function y
reminiscent of similar free-surface conditions
obtained by previous authors (Ogilvie 1968,
Dagn 1975. Newman 1976), by carrying out a
"local linearisation" about the c - 0 solution
z
s(f).
If we substitute appraximatiun (4.5) for
z,($) into the asymptotic recurrence relation
(A.2), we see immediately that this relation
holds if and only if tk(#) is given by (4.2).
t.1,
ABSTRACT
We have utilized the two- And three-dimnsional SOLA codes to investigate nonlinear and
three-dimensional effects influencing the hydrodynamic forces on floating cylinders. In
this paper we discuss nonlinear effects arising
during large amplitude swaying motions of a
two-dimensional 60 triangular cylinder. The
results of the numrical studies are compared
with other data, Including nonlinear potential
flow numerical computations. An interpretation
of the observed nonlinear effects it gi.en. We
present a second study that compares two- and
three-dimensional calculations of the triangularcylinder insway. Here the end effects associated with finite length cylinders are
noted. Nonlinear finite amplitude effects for
thethree-dimensional triangular cylinder are
also considered. To further validate our numerical methods as useful tools for studying
nonlinear flow phenomena, we present the results from a circular cylinder Impacting onto
a water surface.
11. NON.INEAR TWiO-DIMENSIONAL
EFFECTS
I. INTRODUCTION
inthis paper we discuss nuerically determined hydrodynamic forces on floating cylinder%. Particular attention is given to nonlinear effects and to finite length effects. The
nuperical solution algorithms used for these
studies art finite-differance techniques for
the nonlinear Navier-Stokes equations. The
two-dimensional algoritw Is contained in the
SOLA-SURF
code.' This code has been used in
extensive numerical studies of the hdrodymmic forces on rectangular and triangular cylin-
*This work wasperformed jointly under the auspices of the lited State. Energy Research and Develop
ment Adinistration and the Office of Naval Research, ONN Task M11062-465.
38
tangular cylinder and B/T 1.155 for the triangular cylinder, where T is the initial draft
of the cylinder. The amplitudes of motion,
normalized by B, for the rectangular cylinder
were 0.025 and 0.050 and for sway was 0.058.
The triangular cylinder in roll motion rotated
about an axis located at the horizontal center
the wedge and the initial free surface position. The amplitudes of motion were 0.025 and
0.050 radians.
aof
and
y sin
(2)
where a is the amplitude of motion, w is the
frequency of motion, and y Is the anpltude of
the calculated hamonic pressure force on the
body. The phase shift B was obtained by comparing plots of the body displacement and prossure force acting on the cylinder as functions
of time and measuring the shift in phase. A
detailed description of the determination of
these coefficients is given in an earlier report.' The calculated added mass and damping
coefficients for the cylinders in heave, sway,
and rollare shown in Figs. 1, 2, and 3. respectively. The coefficiet are normalized by
PA (pA for roll)and %/B62g, where p is the
fluid density, A is the mean submerged area, B
is the cylinder beam, and 9 is the acceleration
of gravity. Coefficients were calculated for
normalized frequencies (i.e..w v-Mg ranging
from 0.50 to 1.25 with B/T - 2.0 for the rec2.5 -
25
oo0.O A.
v 9K.A-S W C
- Law T"
2.0
1.5 -
1.
a)
.411
0.
co .
oWo 026 o0*
.78
1o
La
.AO
v-.
&a
,
O.1 -the
*
0100
0l0
Fig. 1.
00
020
0.15
WO ISO
Leo
Sincluded
383
*EMP.
eM Mqvul
)
0.20
&0*~J
sSOLAMN
2JO-
OAO
2.0
O-U
J8
ro
E.. 4M (A,0.10)
90LA-uJ (AO01)
LI.W ANNNY
CA.OO0
*- UCLA-WAY
bern
726
1.01
oneI
a
000 0.5
.O01.5
0.5007501.00
00
........
125P
def W0
OAS-theoretical, numerical,
Fig. 2. A comparison of
and experimental values of added mass
(top) and damping (bottom) coefficients
for a 60* triangular cylinder in forced
sway, 'ithB/T1.155.
The added mass coefficient, Eq. 1, for the
larger amplitudes of motion varies less than
10% from the lintr theory values. This follows from its weak dependence on B for small
B.
the linear
These results suggest that
theory reported by Vugts does adequately predict the added mass coefficients for the 60*
triangular cylinder In sway at this bew to
draft ratio and for displacoment amplitudes of
motion up to nearly 50 of its beam width. For
the damping coefficitit, however, this is not
the case. For displacemnt amplitudes greater
inan approximately 25% of the bem width, the
damping coefficient is significantly smaller
thin that predicted by linear theory.
The detailed calculations reveal the reason for the decrersing phase angle B with Increasing amplitude. Referring to Fig. 6, we
set the velocity field of the fluid after 1.48
periods of motion for the four different aplitudes of motion, 0.02. 0.09, 0.12, and 0.16 m.
As the sway asplitude Is Increased, for a given
frequency, the average body velocity Increases.
This causes the fluid to slosh further up
(down) the silas of the body, because surface
waves do not move away fast enough. As a consequence the fluid reaction force on the wedge
teds to be more in phase with the body, i.e.,
the phase shift B is reduced. Of course, this
trend cannot continue indefinitely, because the
Fig. 3.
384
i.i
I1) -
a P hoto
St
# Force A.2
SOLA-SURF, * Duping
a Added Mass
Linear Theory- Dfmp ng
Added lion
- 0.14
-
0.1201U
0.9-
4+
0o~
0a010.
OL-.
0.8Q9-
0.5
00
0.
0
Q4
0OO
IJD
OA--Oi
o4
o.0
On
Ampltude (ON)I
016
Fig. 4. Phase shift end arplitude of the dynamIc pressure force as functions of the
600 triangular cylinder displacement
amplitude.
~
0.6
I
0.04
0&-00
012
0.00
(M)
Amplitude
Fig. 5. Normalized added mass and dampinq coefficients as functions of the 60fl
triangular cylinder displacement amplitude.
case are compensated for by more negative upstream pressures. As a result, the net forces
on the cylinder are nearly the same in the two
cases. The eddies generated in the nonpotentialcase do not carry away kinetic energy because they are alternately generated and destroyed as the body moves to and fro.
III. NONLINEAR THREE-DIMENSIONAL EFFECTS
Nonlinear and finite length effects Influencing the hydrodynamic forces on three-dimensional floating cylinders may be studied
using the SOLA-3D code. We utilized this
three-dimensional code to investigate the end
effects and nonlinear large ami! itude effects
associated with a finite length 60" triangular
cylinder in forced sway.
Finite Length
The parameters for these three-dimensional
calculations were chosen for tomparison with
the two-dimensional calculations. Calculations
were made with sway amplitudes of motion of
0.058 and 0.116 of the triangular cylinder beam
width, i.e..0.02 m and 0.04 m, at the still
water level. The cylinder draft was equal to
0.865 bem widths and the normalized frequency
of motion was 1.25. The cylinder length to
draft ratio was varied from two to four. The
resulting phase shift of the dynamic pressure
force relative to the cylinder displacement
phase, and the amplitude of the hydrodynceic
force per unit length for the thre-dimansional
calculations were virtually the same as the
two-dimensional calculations. This brief study
suggests, therefore, that theend effects of
the triangular cylindr are not significant for
low amplitudes of motion and for cylinder
length to draft aspect ratios greater than two.
Length to draft ratios less than two were not
Investigated.
355
.............
..
...........................
...........
I............
...................
...............
................
.......................
................................................
........................................
............................
777
..........
.. ............
....................
........
---
............
....... ...........
..........
- -------------..........
.................
...................
.
.............
...................
. .......
...................
............
..............
.......
.......
....
..
..............
.....................
..........
..............
.
.................
.........
....
.........................
................
- .......................
..............
........
.....................................
................................
.
..........
.......
-- .....
..............
..........
.........
............
........................................
.....................................................................
........
.............
................
......................
.
.....
...
....
...................
.. ......
....................
.......................................
...................................
..........................................
.....
Fig. 6.
386
fer14 erosfo..
Fig~~~~~~~~~~~~~~~~~~.
eoiy.i...ou..0traglrcyidrinsa
I..
......
.........
u. a..
sne
........
...........
nswydtemne ro
...
6.:::6 rinulrciidr
Fig.~
~ ~ u.........
~~~............
... .r.ote til.....p.on
cod
..........
with .nd.....iu the nonin
I..........
.......
The most significant effect of the increase in amplitude in the two-dimensional clculations, as discussed above, was a significant decrease in the phase shift of the dynamic
pressure force relative to the cylinder displacement phase. The force amplitude increased
linearly with the cylinder displacement amplitude. We made correspondingly large amplitude,
three-dimensional calculations to compare with
the two-dimensional study.
\ N.r./
//
t
,:>,
-ll ..... \\
/.
".V-
\\\ ''\\
..
.....
...
,.... . .li
.... ',-.......
\................................
Fig. 9. Local velocities In planes normil to the ants of the three-dimensional triangular cylinder in
lo amplitude motion after 2.11preriods.
The left plot is the plane nearest the cylinder end
and the right plot is the plane ceadiately outside the cylinder end.
S0
'/
i ,
'
in a vertical plane through tte center of he cylirder and parallel tr.its axis
I,).
1C. lielocities
(left)and in a horizontal Fiae ntAr the vertical center of the cylinder (right) after
2.11 pertoe.s,
The SOI.A-SURF
used tocylinder
calculate
the force
of impact code
on a was
circular
during
constant
velocity
entry
into
a pool of
water. The cyinder boundary was approximated
by straight line segments. The rgd-flu d interface boundary condition applie to each line
segment was successfully used for determining
t
he oa-SR
ces
ue rectangular and
triangular cylinders in forced motion discussed
above. Specifically at the riyid-flu d interface the cell pressure is derived from
ahe
constrain ht the norman fluid velocity be equal
to that of the cylinder. As a free fluad surface approaches a rgi boundary, a simple an-
at
relatively
small ofvelocities,
this ad hoc
linear
combination
boundary conditions
worked very w ell.For the impact problem, however, a modification win necessary beruse the
fluid dd no snticipate the presence of the
rigid boundary in sufficient time before impact
nd the calculation consequently exhibit d un-
300
T1
SOLA-SURF
SOLA-SURF
0.14
1.1 -
SOLA-3D
1.0
0.12
0.10
0.5
0.02
0.00
-0.4
0.12
0.08
0.04
o
---
1,2
* +
:1
o SOLA-SURF
+ SOLA-3D
ery
Lin~
SOS.A-3D
1.2
,
012
S.A -SURIF
-0.--
-----
I
0.08
0.04
0.00
1I.I
1.0
F
Amplitude m)
,.I--
m)
Amrlltuds
1.2
--
[0.6
0.00
o0
0..0
0
0.T
oE
0.04 -
0.9
0.06
ItO.06 -
SOLA-3D
1.1
.. .........
1.0
-.-
+,0
0.0
0.7
0.9
-.
00.5
0.6
0.00
004
0.00
0.00
.12
0.06
0.12
AaPNOWS (0)
Ampltude (M)
Fig. 13.
0.04
Fig. 14.
.. . .......
..
,\-. .
..................
Fig. 15. Local velocities in planes normal to the axis of the triangular cylindcr in large amplitude
motion after 2.0 periods. The left plot Is the plane nearest the cyl;nder end and the right
plot Is the plane imediately outside the cylinder end.
2.0
rids
pe
,I
.
.i
-I
..
_-frequency
for an applied pressure on the fluid just sufficient to bring the normal component of the
fluid and body velocities into agreement at the
time of impact, a boundary condition comination was derived that did force a smooth transition between the free and rigid boundary conditions. The new comlnation uses a quadrntic
in the relative velocity term instead of the
linear term used in the earlier ad hoc expression.
The average pressure on the cylinder, i.e.,
the vertical force per ult length divided by
the cylinder diameter, was determined for a
cylinder with a diameter of 8.25 inches and an
impact velocity of ?.70 ft/sec. The calculation was run to a time of 18.0 msec.0 At this
time the fluid has reached nearly go around
the cylinder. Velocity vector plots In Fig. 18
show the velocity field with the free surface
and the cylinder boundary at -7.85, 2.48, 10.75
and 18.00 msec. Because the calculation starts
some time before the cylinder hits the surface,
we shifted thecalculated time scale so that
the computed and measured peak forces occur at
the !ame time.
that the amplitude of motion be small with respect to the dimensions of the cylinder. Indeed, when this is no longer the case, nonlinear effects, as shown by the SOLA-SURF code.
can be significant.
Three-dimnSional, finite length effects
were determined not to be significant for cylinders with either low or relatively high amplitudes of motion. Apparently the flow around
the cylinder ends, fir the short cylinders
studied, minimizes the pile up of fluid at the
fore and aft cylinder surfaces, which caused
the large amplitude effect in the case of Infinitely long cylinders.
The calculations of the cylinder impacting
onto the free surface forced a needed improvement of tne transition from free to rigid surface boundary conditions. It also served to
further validate the SOLA-SURF code as a useful
toolfor calculating nonlinear fluid flow problems.
10
Re
S .A-mw
..c
...-Ln
t
Fig. 19.
L. I
4
5
3
MINC)
30
......
~~..........................
..
..... . .l l ........
..
, ..
.,,,,
,,,,,
,,,
,.,,,................
..........
.......................
. ..........
:: ::
....
lll
I~
llll
l~ llll
.lll
Jlllll
/ltltll~
lllll
l' l
Fig. 18.
i O i I I~ l l l J J l l l l~l l l l l l l l li
* H* .* *'
....
....
,,, lllllf.............
i i
11iii.1
lllJllll
lli
I.......
illl' llt
l t t l l l ll l
~l i l
Velocity vector plots shewing the velocity field near the irpacting cylinder with the free
surface and cylinder boundary at -7.86,2.48. 10.75. and 18.0 mec.
ACKNOWLEDGMENTS
4.
The authors wish to express their appreciation to Leland Stein for writing the SOLA-3D
code and for many valuable suggestions in running the three-dimensional problem and to
Juanita Salazar for so ably composing the text
and figures of this paper.
S.
6.
7.
8.
REFERENCES
I.
2.
3.
I0
DISCUSSIONS
of four papers
John V. Wehausen
University of Ca~ifornia, Berkeley
I have very few comments on Faltinsen's
paper. Out of curiosity, itmight be interesting to know how big a penalty one pays by replacIng (9)by *(A,y) - 0. Does b(t) have tobc
much larger? It might help some readers if (35)
were identified with the"radiation condition",
At the bottom of p. 9 the author remarks that
C. N.Lee's results do not show a significant
influence of nonlinearity. From hispoint of
view he isprobably right. However, the differEnces
calculated by
Lee, Parissis and by Potash
J. Ship Res.I._5
(1971), 295-324) are significant
Author's
Relt
by OddWN.Faltinsen
It ispossible tu isecentered tiw di ferencing techniques or predictor-corrector techniques that have smaller or no second-order
error. These techniques require more .omputer
time and am.'
unre difficult to implemnnt. Therefore. theinclusion 'if an artific. viscosity
3i
SA
I
a
CONFERENCE PARTICIPANTS
ADEE, BRUCEH., Unive 'ly of Washington, Seattle, Washington, USA
BABAIEIICHI, Mitsubishi HeavyIndustries, Nagasaki, JIapan
BAl. K.J.,DTNSRDC, Bethesda, Maryland, USA
BANNISTER, KENNETH A., Naval Surface Weapons Center, Silver Spring, Maryland, USA
BASTIANON, RICARDO A., instituto tdeTecniologla Naval, Buenos Aires, Argeotina
BORIS, JAY P., U.S. Naval Research Laboratory, Washington, D.C., USA
B#RRESEN, ROLF, The Ship Research Institute of Norway, Trondheim, Norway
BOURIANOFF, GEORGE, Austin Research Aasociates, Inc.. Austin, Teuas, USA
BOYNTON, FREDERICK P., Physical Dynamics Inc., 8erlaley, California, USA
BYERS, DAVID W., Naval Ship Engineering Cent- r, Washilngton, D.C., lUSA
CAGLE, BEN .. Office of Naval Research. Pasadena, California. USA
qALIAL, SANDER Md.,U.S. Naval Academy, Annapolis, Maryland. USA
CHAN, ROBERT K.-C.,JAYCOR, Del Mar, California, LISA
CHANG, Md.S.,DTNSRDC, Bethesda, Maryland, UJSA
CHAPMAN, R. R., Science Applicatilons, Inc.. La Jolla, Califorrnia. USA
CHASZEYKA. MICHAEL A., Office of Navel Retearch, C"hicago, Illinois, USA
CHEN, HSAO.HSIN, American Btbreati of Shippig, Nt
.ork, USA
COLEMAN, RODERICK Md.,DTNSRDC, Bethesda, Maryland, USA
COLLATZ GUENTER, The Hamburg Model Biln, Hamburg, Germany
COOPER, RALPH D., Ottficc of Naval Research, Arlington, Virginia. Ur."A
CORDONNIER, J.-P. V., Unlversil6s de Nantes. Nantes, rrance
CUJMMINS,
WILLIAM F_.DTNSIIDC, eethesda, Mariland, USA
DAOUD, NABIL, University of Michigan, Ann Arbor, Michigan. USA
DAWSON, CHARLES W., DTNSRDC, Betheada, Maryland. USA
!DEMANCHE. JEAN FRANCOIS, Bassly d'Essala des Cardnes, Paris, f-rorce
DERN, J1.C., Bassin d'Essala des Caranes, Paris, France
DOCTORS. LAWRENCE J., DTNSRDC. Bethosda, Maryland USA
EGGERS, KLAUS, Hamburg University, Hamburg, Weal Germany
EUVRARD. DANIEL, ENSTA, Paris, Franca
FALTINSEN. ODD. Norges TelmniakeHogskole, Trondheim. Norway
FEIFEL, WINFRIEL Pd.,The Boeing Comnany, Revlon, Washington, USA
FRITTS, MARTIN J.. Naval ReosearchI .,tb atr Washington, D.C., UISA
GADD,G. L., National Maritime Institute, Feitham. Middlesex, England
OARRISON. C. J., Naval Postgraduate School, Monterey, California, USA
GLEISSNER. GENE H., DTNSF;DC, Bet head,, Maryland, USA
GOODMAN, THEODORE, Stisvens Institute of Technology, Hobokean,Now Jorsey, USA
HAIKOV, ANRI J., BSHC. Verne, Bulgaria
HAUSSLING, HENRY J.. DTNSRDC, Bothea. Maryland, USA
HERMANS, AADJ., Delft ITechnical University, Delft, The Natherravia
HERSHEY. ALLEN V., Naval Surface Weapona Canter, Dahigren, Virginia, USA
HESS JOHN L, John L.Haas Astsociates, Long Beech, California, USA
HIRT, CYRIL W., Loa Alamos Wcentific Laboratory, Loa Alamos, New Mexico USA
HOLT, MAURICE. University of California, Berkeley, California. USA
HONG, YOUNG S., Conaulting NavalArchietr.t Berkeley, California, USA
IIOSODA. RYUSUKE, University of California, Berkeley, California, USA
HSIUNG, CHI.CHAO, Missiasippi Slte, University, Mississippi, USA
HWANG, ALLEN Y.*L, University of California. Berkeley, California, USA
INULTAKAO, U~niversity of Tokyo. Tokyo. Japan
JIACOBSEN, BENT K., Daniah Ship Research Laboratory, Lynigby, Denmark
JIANCJ,CHEN-WEN, Massachusetts Institute of Technology. Cambridge, Massachusetts, USA
391
306
399