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OCR for page 206
Practical COD Applications to Design
of a Wave Cancellation Multihull Ship
Chi Yang [, Fr Inch Noblesse 2, R tina d Lohner ], D tne Hend ix2
~ Instit te for Computatiom~l SCiff es md fommatics
George Mason University, Fci fax VA 22030-4444, USA
2 David Taylor Model Basin, CD-NSWC
9500 MccAr6hur Blvd, West Bedhesdc MD 20817-5700, USA
ASSTRACT
Four methods of 3nshsis c nonlinear method
based on Enler's equations md th e linear potential
flow methods are used to determine She optimal lo-
cation of She outer hulls for c wave carmelhtion mul-
t hull ship that consi ts of c main center hull md two
outer hulls The 6 ee potential -I w methods cone-
spond to c hierarchy of simple approximations based on
the Fourier-Kochin repre sentat i on of ship wave s md She
51ff der-ship approximation
R3TRODUCTION
This study considers m ilhstmtive practical cppli-
cation of CFD tools to c simple ship desig problem
This simple desig case is the wave carmelhtion mul-
t hull ship concept exammed m [1], where experimen-
tcl measurements md theoretical cclcohtions based on
Michell's 6hin-ship approximation are given The wave
carmelhtion multihull ship At is considered consists of
c mom center hull md two identical outer hulls centered
et ( Y.} ~) = (—L Y. =1 I ) with re pect to the center of
the watemhne of She mom hull The main center hull
of the mult hull ship considered m [1] md She present
study hr. c length 2 L'' ~ .390' The main hul l And the
outer hulls are dethred in She Appendix
The study considers the elementary desig prob-
lem of detemminmg the opt mill location of the outer
hulls with respect to She mom center hull, i e She op-
timcl values of the two parameters L Y md Ll, for the
purpose of minimi ing the mare d cg of the ship Four
medhods of analysis are used md compared to one m-
other md to experimental date One of She methods is
the nearfleld -I w cclcohtion medhod presented in [2]
md [3] This method is based on the Euler eq anions
md She nonlinear fre-smface bo mdary condition The
other th ee methods are linear potential -I w methods
that correspond to c hierarchy of simple cpproximc-
tions based on the Fouriff-Kochm representation of ship
waves [ 4] md She s lender- ship cpproxim at i on [5 , 6]
H~1~L(:)~ }i ANDFOURrER-KOCHrN
REPRESENTATION OF WAVE DRAG
Consider c ship advancing along c shaight path,
with const mt speed U. m cc m water of effectively in-
flmite depth Ed lateral extent The flow is observed
from c Cartesi m system of coordinates mm mg with the
ship The V axis is taken along the Pugh of the ship Ed
points t ward She ship b w; i e, the ship advances m
the direction of She positive ~ axis The Z axis is ver-
ticcl Ed points upward, Ed She me m free surface is
the phone Z = 0 The -I w appears steady m the t ms-
hting system of coordinates, Ed consists of the di tur-
barme -I w due to the ship superimposed on c miform
sheam opposing the ship's forward peed The com-
ponents of She distmbarme velocity along She ~ U } Z)
axes are (0~ I .11 ) Thus, She total velocity is given by
(O —U. 1.11 ) Nondimensional coordinates Ed ve
locities are deflmed in temms of c characteristic length
L, takenhere es L = 21', mddhe ship speedUcs
(REV ~)=(U}.ZI/L (tom' ,. ~ = ~-.1 11)/U
Define She Froude m mbff F Ed v as
1~= A/ L " = 1/(2 -)
Here, rJ is She acceleration of g avity
The ding Dl1 = p U2L-C'l1 associated with She
wave energy t msported by the waves hailing She ship
c m be determined from the Hcvelock format
C" = 2r ~ .' 1 k ( 52 _ S: ) (1)
Here, She waken mber /- is deflmed m tffmr of the
Fourier variable 1 by
k(ll) = i,_ i;~ (2a)
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Representative terms from entire chapter:
center hull
Furthffm ore, S' imd,S; iim fhe res I imd imli ginary ps~ts
of the spech m function S = S(`r. 1I Here, 1r is de-
flmed in temm s of the Fourier varii ble ;i by
fr¢;l) = ~7i//
(2f )
7he rellitions (2) foll w from fhe dispersion rektion
F ~3 = f with i = :}~
7he velocity representtition of free-surflice fl ws
imd the rekted Fourier-Kochin representation of ws s
given in [4] deflme fhe spect 3m function 5 in temms of
the velocity dish ibution s t fhe ship hull smfa (or m ore
generally s t s bo mdii y surfti cc fm~t suno mds the ship)
Specificii lly, [4] imd [7] deflme the spfftr m function 5
in term s of s di tr ~buti on of e Iffm entary ws s 1~q ~ [ f ~—
i(
We thus have
5,. - . r = [ (5, )3 - (,5; )- ] - 2 [ ¢5,. )- - (5")3 ]—i.S~
with
,S~ =.I[~:os~¢/~i3)—:/-]
—[.1~t ~ Ob(---r) - .11 ti ~)] <~1~ jli
Here, l, I/trmd lltredeflmedas
-I 1 ((s, )-- (S"''
.t[t ~ = ~ S'',5°-5<'5°
ttJ lS`.S_5~5~'
7hewrr dEsgC'Il ctmthenbeexpressedas
C = C ~ —2 C,, —C r
Here, C4~ rh e 6he
funritions of 6he Fourier varirble jl given by (2) Fur-
themm ore, 6he funrition (-)p is deflmed es
0p = ~—tlil]~( h F. )
(7b)
(9)
with ~ c positive reel con trmt
S mmtrtion in (8) is performed over cll the V
patches ~p u rp that represent the suricce r u r, i e
~ur= ~ ~purp
1~ ~
(9) showr that (-)p ~ U es (~p—4)l(~F I ~—~
7hus, the conh~bohon of c patch ~p to the wrr com-
ponent u I1 is negligible et c point 4 located et c distth ie
(X~F-) rher d ofthe reference pomt l~p cttachedto rp
7hefunritions Ij, Ip, l:^in(8)rhredeflmedas
- Ij = [ (S' _S,, ) c C—(S;—S; ) S5 ] C"'
- [ (5; -.S, ) C'C - (.S': -5,, ),SC ] 5" (10,.)
Ij, = [ (5~ _5; ) c C _ (5~: - ,S': ~ 5C ] 5~'
_ [ (S~ _ S~ ) C C _ (S~ —Sr ) S ] C"~ (l Ob)
Ij;=[~.S':-S' )CC-(S; -~9r ~sc]s~'
_[~5;_Sj)C(_(S':_.S':).S<]C"' (lOc)
6h ~C(] :~9~4tr)1 rmd ~C 41 :~i(~/jl)1
w' i,$C J l mt(4 tr) ~ lS4 J ltilt(7lil) |
Fur6hermore,5' rmdSr in(lo)rhedeflmedas
S., =.Sp''(`r =;1) 5r = SI;(rr =il) (11~)
whffe SI' rmd,S'i rh e 6he reel rmd imr ginary pth ts of the
spech m funrition Sp(`r jl) rhisocir~ted wi6h 6he patch
_p
7 he pectrum functions is given by dish ibutions
of elementary waves over An U Ip Specfficirlh (3c)-
(3c) yield
,Sp = ,SI1:—F:,SI,
,s,, =,4 .t~¢~-). A i,
that defime the location of Che outer-hull his compute-
tiom8 task is darmting usmg typiccl ca Icoktion methods
but c m atua lly be ecsily performed using the slendff-
ship cppr oxim ati on Specificclly, c s mgle - lo op set of
computations of the wave dLcg C'll cssocicted with Che
siff der-ship cpproximation to Che nearfield velocity dis-
tribution et E, md ~O for 4 values of the Froude m m-
ber, 61 values of c md 26values of b, corre pondingto
i x Cl x 2C = C 3ii nearfield fiow cclculations (usmg
11,525 pcnels to cpproximate the th ee hulls) requires
18 hours of CPU time usmg m SGI origm 2000 wiCh
4 processors his cpproah is identified as meChod 3
hereafter
Considerale simplffcations are obtamed ff
nearfield fiow mteractions cre ig ored, i e if the ve-
locity dish~bution et Che center hull ~,. is evaluated for
the center hull clone (i e wiChout the two outer hulls)
md the velocity dishibution et m outer hull is simi-
larly evaluated for Che outer hull clone (i e without Che
centff hull md the oCher outer hull) his cpproximc-
tion, which aco mts for farfield wa~ interference ef-
fects but neglects nearfield fi w interations, only re-
quires two nearfield fi w evaluations (one for the cen-
ter hull, md one for m outer hull) per Froude m mber,
c task that c m ecs ily be perfomm e d using exist mg ccl -
culation meChods he pectrum imctions S'' md S°
defimedby(3) mdCherektedfunctions I, 1/t md 11
given by (5) simibrly need be evaluated only once per
Froude n mber hese pectrum functions, lik Che ve-
locity di tr~butions et Che th ee hulls, are independent
of Che parameters a md h, which only cppear m (6d),
withm Che "negligible nearfield interation" cpproximc-
tion hus, Chis cpproximation effectively uncouples
the outer-hull location pcMmeters (a ~ hl md nearfield
fiow cclculati ons For the p mp o se of est mm ating the im -
port mce of nearfield fi w interations upon the wave
dLcg <.1t, the nearfield velocity dish~bution is evalu-
cted here using the slender-ship cpproximation clrecdy
used in method 3 his cpproah is identff cd es meChod
2 hereafter hus, comparison of meChods 2 md 3 pro-
vides msight mto the import mce of nearfield fiow intff-
ation effects Method 2 conesponds to Che fikst-order
siff der-ship cpproximation defimed in [5]
Insight into Che importmce of usmg c sophisti-
cated nearfield fi w calcoktion method c m be gained
by comparmg method 2 md meChod I, which cone-
sponds to Che :D:roCh-order slender-ship cpproximation
in [5] md to Che trivicl cpproximation
I = 1( 1' = (~t r) - 15
in (3d) md (3e) hus, no nearfield fi w cclcoktion
is required m this simplest cpproximation ~deed, Che
:D:roCh-order slender-ship cpproximation is cssocicted
with Che hivicl cpproximations t x rt = 0 md t 7 = 0
for the t mgenticl components of the velocity t et Che
ship hull ~ mdwaterline F. es previouslyexpkined
Methods 1, 2, md 3 are based on the Fourier-
Kochm representation of ship waves, i e on linear
potenticl fiow, md the imthff simplffcation cssoci-
cted wiCh the slender- ship a pr oximat ion hus , ew~n
meChod 3 only aco mts for effects of nearfield fiow
interations in m cpproximate marmer H wever,
nearfield fiow interations are fully taken into aco mt
in the nearfield fi w cclcohtion method presented m
[2] md [3], which is based on the Enler equations md
the nonlmear free-smfa bo mdary condition his
meChod is identified es meChod 4
RESULTS OF ANALYSIS
Figure I depicts Che experimental values of the
residuary dLcg coeffcient C'lt given in [1] md the
corresponding predictions of Che wave dLcg coeff-
cient C'll given by methods 1, 2, 3, md 4 for
four arragements of the outer hulls hese four ar-
r mgements of Che outer hulls correspond to a =
—C. 128 —0.203 ~—0.2 iC —0..38 i md h = 01 3C (Che
same value for cll four cases) he left col mn m Fig
I comparesC"t mdC'Il predictedbymeChods I md2
herightcol mnshowsC'll mdC'Il givenbymeChods
2 md 3, mdbymethod 4ct F = 0.20.0.3.0.i.0.i
D fferff es betweff the values of C'll predicted by
meChods I md 2 Jefi col mn) mdbetw en meChods 2
md 3 (right col mn) are iri Iy smell, md these 3 meCh-
ods yield values of C'll that are in fai C8 eement with
the experimental values of C'lt In particular, the vari-
ction of C'lt with respect to Che Froude m mber F is
well ccptured by Che Cheory he values of C' 11 given by
meChod 4 et F = C.2~,C.3 0.i,0. 3 cre in evenbetter
C8 effment wiCh C'll on the whole, md are in irirly good
C8 effment with C'll predicted by methods 2 md 3
Figs 2a,b,c,d compare the values of C' 11 given by
meChods 1, 2, 3 for F = 0.3.0.i.0.3.0.2iad outer-
hull arrmgements withm the region
—0.7i~a<0.7~ 0.C0
outer-hull k rkmgementg are indicated m red Figs 2
indicate that methods 1, 2, kmd 3 predict be t (blue re-
gions) kmd worst (red regions) outer-hull k rkmgffmentr
theta e m tong good cement
Fig 3 sh ws the values of the WE d kg coefifi-
cient C'il given by methods 1, 2, 3, 4 at the 6 ee be t
distinct outff-hull knkmgementg i e for (ak be I with
k = 1~2 ~~pred~c~edh~ hod ' 7hisfigmeshows
that, c16ho rgh the values of C'il predicted by methods
1, 2, 3, 4 k e not identical, These four methods yield
C'l' < C"< < C"< The remits depicted in Figs I, 2,
kmd 3 indicate thm methods I kmd 2, which are com
putatimurlly m ore efficient th m methods 3 kmd 4, may
be used for The p mose of determining optima I k
outer-hu used for ire p mom of dtle~minmg op
ther st dy the best kmd worst outer-hull k rkmgementr
Fig 4 depicts the variation of the WE d kg co-
efificient C'il, for F = 0.3 kmd F = 0.,, withm the
region
-1<~<3 0<1A<1
Conridcrktion of this large region, which g early ex-
tends The region (16) of practical mterest for The de-
sig of k WE Ck ellation multihull ship, provides k
broader View of the variation of The WE d kg coefifi-
cient C'i1 withm The smaller region (16) exammed m
Figs 53.b
Figs Sa,b sh w the variation of the WE d kg co-
efificient C al predicted by medhod 2 within the region
(16) with For = 0.02~ kmd ~1~ = 0.01, as in Figs 2a-
d, for 20 values of The Froude m mber F m The
0.21i7 < F < 0mi2C Thus, Figs Sub present the
result of C1 x 2C x 2U = 31, 720 An 31b3tions of c it
As in Figs 2,1 we t kmdhighert values of C", i e be t
kmd worst outer-hull managements, core pond to blue
kmd red regions in Figs S
The left side of Fig 6 depicts the variation, with
rerpfft to The Froude m mber F. of The WE d kg co-
efificient C,11 _ 2 C Al kmd of the WE d kg coefficients
Catty, kmd Pilots corresponding to The best kmd worst
outer-hull k rkmgementr fo md (using method 2) within
the region
-0.7J<~<0.7J U.l
the present study) of comparmg md validating zltenuu
tive methods of zrurlysis of main intere t m fhe present
study Fmthemmore, conshamts zssociated with mis-
sion requirements, tructural considerations, seakeep-
ing, md course keepmg must evidently be considered
The th ee medhods based on the Fourier-Kochin
representation of ship waves md the rlff der-ship zp-
proximation, especially the medhods (called methods I
md 2) corresponding to the :D:roth-order md flk t-order
rlff der-ship zpproximations given in [5], provide sim-
ple md highly eff cient tools These practical tools have
been sh wn to be zdequate for the purpose of deter-
mming optimal locations of the outer hulls Medhod
4, based on z more reflmed fl w am~lysis, c m then be
used effectively to fmther evaluate the flow zt fhe opti-
mal outff-hull arr mgement
The use of z pragmatic zpproach that relies on z
combination of simple md more reflmed tools evidently
is z well~stablished practice in particular, the :D:rodh-
order slendff-ship zpproximation (i e method 1) had
previously been used with success for hull-fomm opti-
mization m [10] md [11] Thus, the practical useful-
ness of this remarkably simple zpproximation is con-
flkmed in fhe present study
ACKNOWLEDGEMEN S
The work of Ymg md Lohmer was partially
f mdedby AFOSR Dr L onidas Szkell techmical moni-
tor) mdbyNtLLCP&FD DrWillizmSmdbergtffh-
nical monitor) The work of Noblesse md Hendki was
supportedbythe LRprogamatNSWC-CD Allcom-
puter ms were perfommed on z 128-Processor R10000
SGI Origin 2000 zt the Na~l R search Lzboratory
3EFE 3ENCES
[1] MB Wilson, C C Hsu & D S. Je kins (1993)
Ezpezim mt and predictions of 6he zesutance chazac
tezutcs of o w ve cancellohon multibull ship concept,
23rdAmeri mTowingTmkCo f,l03-112
[2] R. Lohmer, C Ymg, E Oime & S. Idelssohm
(1999) An unst~tuxed grid bosed purollel free
suzioce solvez, Applied N merical Mzthffmatics 31:
271-293
[3] C Ymg&R Lohmer(1998)Fulynonlineorship
wow colculohon using unst~tuxed grid and purollel
compuhng, 3rd Osakz Coll Advarmed CFD Applicz-
tions to Ship Flow md Hull Form Desig, Osakz, 125-
150
[4] F. Noblesse (2000) Velocih Zffp~ mtohon of
f zee~uzioce f w and Fouziez Koch in zepres m t tion of
wo~, submitted
[5] F. Noblesse (1983) A slendemship 6heo y of wove
zesutance, Jl Ship Research 27: 13-33
[6] F. Noblesse & G Tri mtafyllou (1983) Ezplicit
opp?ozimohonz fw colculohng p t ntiol f w obout o
body, n Ship R search 27: 1-12
[7] C Y mg, R. Lohmer & F. Nob le s se (2000) Fozfie d
extension of nff zfield steody ship w ves, Ship Techmol-
ogyR search47: 22-34
[8] F. Noblesse, XB Chen & C Ymg (1999)
Genezic supez G?e~m funchons, Ship Techmology R -
search 46: 81-92
[9] F. Noblesse, C Y mg & D Hendkix (2000) Steody
free~uzioce p t ntiolf w due to o p intsource, 15th
Intl Workshop on Wzter Waves & Floating Bodies, is-
rael
[10] JS LetcherJr,JK Marshall,JC Oliverlll&
Nils Szlvesen (1987) Stom d St ipes, Scientiflc Amffi-
c m, 257: 34-40
[II]DC Wyatt&PA Chmg(1994)Developmmtond
ossessment of o tot I zesutance optimized bow foz the
AE 36, Marme T - hmology 31: 149-160
APPENDIX
The outer hulls are considered flrst L t 2L°.
2B°, md D° t md for the lengh, beam, md dLaft of
m outer shut, which consists of z parallel m idbody md
parabolic sharp~nded nose md tail regions Deflne z
system of coordi mes (A-° } ~". Z°) with origin zt 6he
center of 6he outer trut The upper part of the shut is
givenby } ~° = = L<'wi6h B<' deflmed zs
L < V° ~ Y/:
( ~ i-° - i-/i )') for A-// ~ A-° ~ L°
Lt = L (l (L°—Y<'j ) for —L° ~ v° ~ ~ "
These equations hold for—D°—Lt' ~ Z° ~ O The
I wer part of the trut is given by
} ~ = =B<' th~9 Z =—D°—Lt'(l—ctw9)
with U ~ 9 <, /2 Thus, the outer hulls zre deflmed by
the fl e pzMmeters L°, G°, D°, i-f; md V~4
The centers of the waterpkmes of 6he two outer
shuts are located zt A- =—L Y md } - = =LI wi6h re-
spect to the center of 6he waterplane of the mam cen-
ter hull D flme 6he ystem of coordim~tes (A-~}.Z)
with origm zt 6he center of 6he waterpkme of 6he cen-
ter hull Thus, 6he coordinates ~ V.} - Z) corre ponding
to 6he outff-strut coordinates (A-° Y ° Z°) zre givenby
V=A-°—LX }-=}-°=Li Z=zo
for 6he outer shuts centered zt ~—LX =Ll I The
outer-strut smfaces c m be deflmed m temms of the
centff-hull coordinates vie the foregoing coordinate
t msfommations
he main center hull is now considered Let 2 L''.
ELF, md D'' t md for the length, beam, md d aft of
the center hull he local beam (wide t waterline) of
the center hull is given by } ~ = = L ''' with L ''' defmed
as
L,,, = L' for ,V,'; < ,X' < .~~1;
Be = B (1—, L — lily) for it; ~ ~ ff L'
L s ~ = L' (1 - (LY ';; ~ ~ ) for —L'' < ,\~ < I'm
he top waterline of The center hull is given bv
} ~ = = Lu with Lo defmed es
Bo = Lo for ,V,'; ~ .\' ~ ~ ;.
( (L —.\ /; ) ~ )
Bu = Lo (1 - (Lo _'~',;)~) for —L: < ,~' < .;';
hus, The local beam md the top waterline of the mom
center hull are defmed by the si parameters 1'', D'',
L', Lo, ,X'/: md An;:
Every tramebne consists of c straight horizontal
bottom md c straight But not vertical) side com cted
by c portion of circle of radius equal to curbs,, where
L,,, is the local beam md o = 1/2 hus, The hull
bottom is defmedby
Z=—D'' for —(1—~ ~ L,,, A: } ~ ~ ~~—~IL'''
he hull side is given by
}-==(Lo—C'Z) for —Dy
Framelines are vertical at some distance above the
waterplane Z = 0. The straight hull side below the wa-
terplane, inclined at an angle A* = tan-i r with re-
spect to the vertical, and the vertical upper hull are con-
nected by a circular arc of radius R. Consider the por-
tion of a fralnelille located in the half plane Y > 0.
The frameline intersects the waterplane at the point
{Y = Bo, Z = 0) . The center of the circular arc that
joins the lower and upper framelines is located at the
point
Y = Bo + R cost* Z = R sing*
The contact point between the upper vertical frameline
and the circular transitional portion of the frameline is
Y =Bo—R{1—cost :! Z=Rsin3*
Thus, the center hull above the waterplane Z = 0 is de-
fined by
T~=~:Bo—R{1 - 1/\/~)]
for RT/~ < Z
The circular transition region between the upper verti-
cal frameline and the hull side below the waterplane is
defined by
() =~[Bo—R{cos9—1/~)]]
l Z = R { rat—Silly ) J?
for 0<9<9*=tan~~r
The relations cosp* = 1/~, Sill3* = r/~
were used above. The radius R may be taken propor-
tional to the beam 2 BoC of the top waterline, i.e.
R=)Bo
with ~ = 1.
The six parameters LC, DC, BC, BoC, -YB and _Ys
that define the main center hull and the five parameters
L', By, D-, -YB and _Ys. that define the outer hulls are
given by
Lc=0 5 DC ~ o 0,-49 Bc~ 0.0245
B,2 ~ 0.0128 ~ ~ 0.105 _Y,sC ~ 0.0401
L~~ 0.2436 D°~ 0.0356 B°~ 0.00847
YE ~ 0.175 As ~ -0.134
for the wave cancellation multihull ship considered in
[1] and the present study The experimental results
given in [1] and reported in Fig. I of this study cor-
respond to four locations of the outer hulls given by
{—LX,lL9) with
Lx ~ o 128,0.205,0.256,0.385 and Lo ~ 0.136
Wave cancellation multihull ship model
Cw metb 6 2—
Cw metb 6 ~— t
C nFelmept a ~ I
1~ f?
uu 2 0 ZS
03 03s 0; 045 05
Fsu6esum6elF-0~26'
uu 2 0 ZS
03 03s 04 045 05
FxudesumbelF-020s)
00024
0 002
00016 .
00012 ~ t ~t t
00006
00004 .
uu 2 0 ZS
055
03 03s 0; 045
FxudesumbelF-02s6)
00024 _
0 002 .
00016 .
00012 .
00006 .
00004
O-
02 025 03
055 U2
l
05 055 UZ
00024 _
0002 .
00016
00012
00006
00004
03s 04 045 05 055 uu Z
FxudesumbelF-036s)
C metb 6 2—
C metb 63—
C metb 6 4 s
ClexF Imest m
Ozs 03 03s 04 045 05 055
Fmdesumbella-0126)
uz Ozs 03 03s 04
045 05 055
~ti~jf ~
Ozs 03 03s 04 045 05 055
Fmde sumbelfa-0256)
~ r ~ ~ ~ ~ r ~
'& '
025 03 03s 04 045
Fmde sumbelfa-0365)
Fig I Cmicohted wxwe d mg and experi ental residuary d mg
Cw method 1
Cw method 2
Cw method 3
Fig. 2a. Wave drag coefficient predicted by methods 1,2,3 for F=0.5
Cw method 1
Cw method 3
7'
Fig. 2b. Wave drag coefficient predicted by methods 1,2,3 for F=0.4
into ~~.
. 6~1~4 ~ ~
Cw method 1
Fig. 2c. Wave drag coefficient predicted by methods 1,2,3 for F=0.3
Cw method 1
Cw method 2
Cw method 3
Fig. 2d. Wave drag coefficient predicted by methods 1,2,3 for F=0.25
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
n I
Cw method 1
Cw method 2 ----~----
Cw method 3 -----~-----
Cw method 4 ----------~----------
Cw method 1
Cw method 2 ----~----
Cw method 3 -----~-----
Cw method 4 ----------~----------
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
o
0.001 2
0.001
0.0008
0.0006
0.0004
0.0002
O-
2 3 1
F=0.40
Fig. 3. Wave drag coefficient at three best arrangements of outer hulls
Cw method 1
Cw method 2 -----~----
Cw method 3 -----~-----
Cw method 4 ----------~----------
... -.~3-----
Cw method 1
Cw method 2 ----~----
Cw method 3 -----~-----
Cw method 4 ----------13----------
2
F=0.50
3
F=0.5
Fig. 4. Wave drag coefficient for—3 < a < 3 and 0 < b < 1
Fig. 5a. Wave drag coefficient for 10 Froude numbers
F=0.5124
1 F=0.4521
F=0.4220
F=0.331 6
F=0.31 65
Fig. 5b. Wave drag coefficient for 10 Froude numbers
F=0.2713
F=0.2524
F=0.2449
1 F=0.2374
F=0.2223
1 F=0.2147
Cwwoff amangeme~s _
Cwbe~ ng~eme~
:,~'1
C wwst a~mngeme~s _
- C best a~mngeme~
Fig 6 Wa~e d cg for best md worst arr mgffmenb of OOtff holls
. . .
1~| 1!1 ~
0 6 ~ ~ ~ ~ ~
02 025 03 035 04 045 05 055
F~oude numbe~
026
026
O022
016
016
Oo112F_ , ' 1
02 025 03 035 04 045 05 055
F~oude numbe~
0 0024
0 002
0 0016
0 0012
0 0008
0 0004
o
Cr expenments a= 0128
Cr expenments a= 0 205
Cr expenments a= 0 256
Cr expenments a= 0 385
Cw near best arrangement
Cw best arrangement
x ~ ~ t ~ x~
^ ~ it
, ~ ~ ~ ~
7 ~ ~ o . ti . ~ B '
! ~ tY tt ~
—~
''' ~ tK
02 025 03 035 04 045 05 055
Froude number
Fig 7 Wave d cg md parameters c md b for best md nearbest arr mgements of ooter holls
DISCUSSION
L J Doctors
University of New South Wholes
A shalic
I would like to express my appreciation to the four
abhors for c most interesting pap r on the subject
of multihulls, c matter of mte~est to m my
researchers who are aiming to reduce either the
overall resistance or She wave resistance (es in the
present work)
The contour plots of Figure 2, Figure 4, Ed Figure
5 me m excellent way of presenting the
considerable quantity of date showing the
i fluence of longitudinal stagger Ed lateral offset
onthe wave resist mce
Could She mthors kindly explain the background
behind the breakdown of the Green f notion mto
She f ee terms m Equation (14)? The first two
terms are w 11 k ow Ed the (new) third term
provides the correct limiting behavior for low Ed
high Froude mmmbers However, there seems to be
no evidence of wave-like behavior m the third
term b deed, there would be c family of similar
versions of the third term that would e habit the
appropriate limitingbehavior
Finally, c m the mfhors indicate whether the
effects of sirJcage Ed trim are included m their
work Ed whether they feel These would effect the
find predictions for the wave resist mce?
AUTHOR'S REPLY
Thank you for your interest in our work Ed your
q estions The fl w field Ed the Green function
are expressed es sum of c wave component Ed c
local component, es is indicated m equations (7 i)
Ed (7b) Equation (14) provides m extremely
simple approximation to She local component
alone, not She wave component This component
is defined by equations (3) f ough (I M-d) with
expressions (3c<), or expression (13) in the
slender -ship cppr oxim it ion
E fects of sirJcage cod trim have not been included
m our calculations, Although this could be done
without my greet difhculty Our guess is that
sirJcage Ed him would not sig if icmtly modify
She optimal hull ant mgement, but it would be
Interesting to verify that this is indeed the case
As noted in our reply to the discussion by Frofs
Nakatake Ed Ando, Ifftmg effects are not Included
m methods 1-3 (thus, only sources me used in
These 3 potenticl-flow methods) but are taken mto
account in medhod 4
DISCUSSION
K Nckatake Ed J. Ando
Ky shu University, Jcp m
We con rat Ate you on your paper to predict She
optimal location of the outer hulls for She mom hull
by applying the rather simple wave d cg formulas
We also applied our k mkme source method to c
trimar m Ed co firmed by some experiments that
the total wave d cg fairly ch m e. according to She
location of the outer hulls for He main hull From
Fig 4, it is interesting to note Nat the wave d cg
coefhcient ch m e. like c wave contour
Among your methods, method 4 seems to be most
exact in She strict sense, the main hull m be
treated es c nonlifting body by method 4, but He
outer hull should be heated es c lifting body
bee mse the flow around it is not .-.~nm en iccl with
respect to the center plane Therefore the
cdditiorurl vortex (or doublet) dish~bution is
needed to satisfy She Kutta's condition et He
tnailmg edge of the outer hull This effect may
become k ger when the h msverse distance
betw en She main hull Ed She outer hull is smell
Whet do you thi k of this point?
AUTHOR'S REPLY
Thmk you for your interest in our paper Ed for
providing Information cutout your own work on She
effect of hull arr in emenr upon wave dmg We
agree Nat the outer hulls should m principle be
treated es lifting surfaces, Ed that lifting effects
c m be expected to be larger if She outer hulls are
closer to the center hull The 3 simple potenticl-
flow methods w have used (methods 1, 2, 3) do
not include lifting effects Medhod 4, based on She
Euler equations, accounts for lid ing effects The
restively good agreement between experimental
residuary d cg Ed wave d cg predicted by all 4
methods show in Fig I sugge ts that Ifftmg
effects may not be very impo t mt in the present
case
