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OCR for page 553
Numerical Prediction of Scale Effects in Ship Stern Flows with
Eddy-Viscosity Turbulence Models
L. Eca ( nstituto Superior Tecnico, Portugal)
M. Hoekstra (Maritrne Research Institute, Netherlands)
Abstract
his pcper presents c m mericcl mve tigation of
sccle effects on ship tem flows he mo t popu-
kr cigebrcic, one-equation md two-equation eddy-
viscosity turbulence models are successfully cpplied
to fhe calcoktion of fhe flow aro md fhe Mystery
tmkerfiom modelup to full sccle R y oldsmmber
It is show that fhe choice of the tmbulence model
does have m i fluence m fhe results (notably fhe
near wake fleld), but the dffferences c msed by ch mg-
mg fhe turbulence model tff~d to dim mish wifh the m-
crecse of th Rey olds m mber
1 Introducidon
One of the mcin cdvmtages of Computatiorurl
Fhid Dynamics (CF ) over haditiork~l model testmg
is the potenticl) c mctility to p~edict R y olds m m-
ber effects on fhe flow fleld aro md c ship But be-
fore this cdvmtage cmbe exploited, two major tasks
arise he fl st is to chmge th potenticl ccpability
to c true ccpability by mcking sme fnat fhe m mer-
iccl method c m cope with the ext~eme reqmir ments
posedbyfl wsimohtionsatfullscaleR y oldsmm-
ber he second task is to go th ough fhe verffcation
md validation processes
Only c few ctt mpts to p~edict scale effects with
CF have been reported One such cttempt is [I ], p~e-
sentmg ~esults, h wever, which are not m mericcily
convmcing he present cubhors have show to be
more succes dul in computmg ship stern fl ws from
model up to full sccle R y olds n mbers, [2], with
ver iflca t ion of m m ericcl error s, [3] md [ 4] U for -
tmutely, there me vi tually no ~elictle experimentcl
dch a~ihtle for full scale ship tem flows, so that
fhe validation process is obsh ucted he best fhing to
do is thff~ to inxecse fhe level of co fldence by vali-
datmg et model-sccle R y olds m mber md sh wmg
fnatw 11-kmow t~endsforR inxecsmgaresytem-
ctically reproduced
~ the hst two decad s, chuge efforthcs been mcde
to validate CF predictions et model sccle R y olds
n mber H wever, the present tat s is far from be-
mg completely satisfactory Notably fhe accurcte p~e-
diction of the axicl velocity fleld in regions of high
treamwise vorticity hcs proved to be diff cult Some
success hcs been ckimed for second moment clo-
sures, e g [5] But if one cims et selecting c tmbu-
lence closure that is m merically robust et model and
f 11 sccle R y olds n mbers, eddy-viscosity models
are still fhe only ~ecsoruible choice
~ fhis p mer we present c m mericcl inve tigation
mto fhe prediction of sccle effects with eddy-viscosity
tmbulence models, mcludmg cigeb~aic, one~quation
md two-equation models Two mcin gocis are con-
side~ed:
e Ir~stigate which turbulence m odels are m meri-
ccily robu t from model up to f 11 sccle R y olds
m mbers, without fhe need of fmfher t ming
e Evalucte fhe i fluence of fhe R y olds n mber
on fhe diffe~ences betw en sohtions obtamed
with dffferent eddy-viscosity tmbulence models
With cchie ing these gocis, we expect to inxecse the
co fldence m fhe use of CF et full sccle R y olds
n mbers, usmg turbulence models that have been
origincily d veloped for thin shear hyers et model
sccle Rey olds m mbers
The pcper is orgmised in fhe following way: sec-
tion 2 gives c brief description of fhe tmbulence mod-
els md fheir m mericcl impleme htion The ~esults
of cpplication to fhe fl w aro md the Myste y t mker
ctR y oldsn mbersfiommodelsccleuptofullsccle
Rn me presented md discussed in section 3 Section
4 summarizes fhe conchsions of fhe pcper
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2 Turbulence Models
~ order to cover ~ broad spech m of eddy-
viscosity tmbuler e models, w bav considered 6be
following models:
e Algebraicmodels
Cebeci & Smith, [6, (CS)
Baldwin & Lomax [7], (BL)
e Or ~quationmodels
Spalart & Allmaras, [8], (SA)
Menter, [9], (MT
k—d models
* Two-kyer model, [10], (KE-TL)
* Cbien's low-Rn model, [11], (KE
k—m mod is
* Stmdard, [12], (KV~
* M nter, [13], (KVvM)
q—5 model, [14], (Q~
Tbe or -eq wtion model of Baldwin & Bardb, [15],
md 6be SST v sion of M nter's k—m model, [13],
w re aIso tested How v r, remlts of 6bese models
will notbe mcluded indbe presentpaper Tbe Baldwm
& Barth model sh wed ~ v ry poor bebaviour m 6be
mitial te t r ms, while the SST v sion of Menter's
k_ m does r t perform berer 6 m 6be other k_ m
models te ted
It is poss~ble to improv 6be quality of the pr dic-
tions of ship tern flows wibb the or -equation md
two-equation tmbuler e models usmg ~ simple cor-
r ction to the production term of 6be hansport equa
tions, [16] However, in 6bis paper w will gdopt 6be
tmdardv rsionsofthemodels
2.1 Algebraic Models
Tbe two algcbrdic models are w 11-kmown md
based m ~ two-hyer d flmition of the eddy viscosity,
v~, where 6be eddy viscosity is obtamed from 6be m m-
im m of its values m 6be two 1ayers ~ 6be i mer-
kyer, bobb models use 6be mixmg-length approach
wibb the Vm D lest dgmping f mction m 6be r ar-wall
rgim
~ 6be Cebeci & Smithmodel, the eddy viscosity in
6be outer region is obtair d from
(v,)O = 0.0168q~6~ j (1)
wher q~ stmds for tb v locity at 6be edge of the
viscous region md 6* is the dispkcement thick-
r ss, which is m integ al parameter deflmed for 2-D
bo mdary-hyer fl ws, md ~ is the mtemmitter y fac-
tor, which is giv nby
~ =
2)
wher 6 is 6be thickmess of tb viscous region
~ ~ ship stem flow calcohtion, perfommed m
curvilirRar coordir~te system, (; ~ 5)l some as-
sumptions bav to be made to compute q~, 6* md ~
We detemm me these q mtities fi om i fommation along
each g id lir normal to 6be wall md the viscous 1ayer
6bickmessiscalcoktedfr mbbetotathead D tailson
6be calcoktion of q~, 6* md ~ me giv n m [16]
Tbe main advmtage of 6be Baldwin & Lomax
model ov r 6be Cebeci & Smibb model is the abser e
of 6 md 6* fiom the deflmition of 6be lengh scale
m 6be outer r gion in 6be Baldwm & Lomax model,
(v,jO is giv nby
wher
(V~>o = 0-ol68ccpl;~cledFw te i
(3)
Fwww = mm (ymwFmwicCud If (4)
Fmw is 6be maxim m of the f mction
F=yrlwl(l—e~~ ) (5)
md y w is 6be value of Yr where F w occurs Ud~f is
6be differer e betw en the maxim m md mmim m
values of q along m ~ g id lir, I Wl is 6be magmitude
of 6be v rticity v ctor, A = 26 md Yr is 6be non-
dimff~siorul di tance to 6be wall in wall coordir~tes
Ftl~d is 6be equivalent of ~, md is giv by
Ftied ~ 6 (6)
+ ~ tledymw)
Ccp = 1.6, Cg = 0 25 md Ctl~d = 0-3
Althoughtb calcuhtionofFww~ mdymw seemsto
be stmighfforward, it is not Tbe values of these two
parameters me directly rehted to 6be v rticity, which
ismv selyproportiorultotheg idlmedi tanceinthe
phy ical space Tbis depender y of F on 6be v rticity
makes its m merical calcuhtion extr mely sensitiv to
'4t sstemwisecoordinde.~t sooordt decorm~lolhe
hipsmbace md~t sgtdhwisecoordinde
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Representative terms from entire chapter:
turbulence models
6he me m flow field, which dep nds on v~ md is d -
termin d itemtiv Iy To cvoid umdershoots md ov -
shoots of the eddy viscosity, the spaticl variction of
Y~~m; must be limited in 6he present implemenbtion,
6he following limiters are cdopted
0.95y,,m,
2.2.3 Boundary Condfldons
~ c ship tem flow cclculation we n cd to spec-
ffy boumdary conditions et 6he si boumdaries of 6he
domcm: mlet, outlet, ship su face, ship symme-
hy pkme, fiee suface: md external b mdary The
boumdary conditi ms are specffed in the wme way
for both on ~quation tu bulence m odels At the ship
surfae the tmbulent qumtities are vmo md symme-
hy conditions are cpplied on 6he flee su face md et
6he ship s symmeby plane At the outlet, the sheam-
wise derivativ of 6he tu bule t qumtities is cssumed
tobe:owo Di ichletboumdaryconditionsme imposed
for the tu bulent qumtities on 6he external boumdary
md are fl cd by 6he valu on 6he border of the inlet
pkme The p~esent calcoktions w re pe formed only
for 6he di p ut of the ship; 6he mlet plane is located et
mid-ship, md so there is cl ecdy c viscous ~egion in
6he inlet pkme A t mdard st~aighffo ward procedme
was cdopted to deflme the tmbulent qumtities et 6he
mlet plane: v~ is ccl ulated with 6he Cebeci & Smi6h
algeb~aic model md the tmbuient qumtities are 6hen
deriv d fi om the km wn eddy-viscosity
2.3 Two-equation Models
2.3.1 Two-layer k—~
The two-kyer k—d model p~esented by Chen
md Pctel m [10] solv s two h m port equstiom in
6he outer flow region ~ 6he n ar-wall region only
6he equation fcr 6he tmbulen e kmetic en rgy, k, is
solv d The valu of d is deriv d from m clgebrcic
length sccle The n ar-wall model is equivalent to 6he
on -equationmodel of Wolishtein, [17]
The eddy-viscosity is obtain d fi om
V, = C~—
(I 5)
md k md d for c stecdy flow me obtain d from 6he
solution of 6he equations:
=v~S+V ((v+—)Vk)—~ (16)
md
=C~kv~S+V I~,l~,v+—JVeJ—C k (17)
the n ar wall region, d is detemmmed from
kl i
c=0.416 / j
(l 8)
slmbe presecl o~ ml dioc we haw m ed Ihe double model m
prox m diom md so Ihe tree whace is ssymmet y~l me
wi6h
md
/e =Yr (1—e—o 263de~) (19)
Rq = Y
20)
The f~ f mction deflmed by the Woffshtem on -
equation model is
I—e—O.Ol de~
f~ = I _~0.~63de,
21)
The stmdard k—d constmts me cp = 0.09, Cl =
1.44, C: = 1.92, t = I md t = 1.3
The k y fectme of 6his two-kyer m odel is the deter-
minati m oftheboumdarybetwe nthe im~er md outer
kyers which is often deflmed by c criteri m based on
y However, withy it is diflflcult to establish c cri-
terion which is insensitiv to 6he Rey olds m mber in
ou cpprocch, 6he i mer-lcyer region is deflmed by the
following criteric:
f~ < 0.99 Ay < 50 .
The fl t criterion would be 6he nstmal choice to bor-
der 6he im r-hyer ~egion How v r, in the iterctiv
detemmination of the eddy viscosity fleld it may lecd
to excessiv Iy large regions, which provoke m meri-
calconvergeneproblems Therefore,w hav cdded
6he second criterion which originates fi om the kmowl-
edge on flct pkte boumdary kyers, where the fully-
tmbuient region starts et y ~ 30 - 50 This cp-
procch does not guarmtee that 6he f~ is close to I et
6he edge of im~er-kye' Therefore, m 6he outer-hyer
6he d h m port equation is solv d but f~ is still ob-
tain d from (21)
As in 6he one-equation models, iyr>~. deflmed by
equation (11 ), is used in the wake to ~epresent the dis-
tmce to the wall
2.3.2 Ctden's k—d model
The low R y olds k—d model proposed by Chien,
[11], does not distingmish betw en im r md outer
kyers md is directly cpplicable in 6he n ar-wall ~e-
gion The eddy viscosity is obtamed from equation
(15) The k md d h m port equations of this model
D k v S + V ( (v + vt ) Vk) _ _ 2 k 22)
md
~f = C, kV,S—k ~C2fi~ + T] +
V ((v+ vq,}V~) .
7he near wall dampmg f mctions are giv n by:
f~ = I _ e_c~; j
7he model constants are
t=l—I 8e {~~) . 25)
cp = 0.09 j Cl = 1.35 j C: = 1.8
t =1 j t =1.3
C3=0.0115 j C4=0.5 .
2.3.3 Standard k—m model
7he k—m model hcs been proposed by Wilcox,
[12] it obtams 6he eddy-viscosity from
k
V, =—
md k md m are obtsined fiom 6he solution of 6he
equations:
giv by (27) 7he m t msport equation is re-w itten
as :
¢= ~S+2(1—Fl)(q~:Q~Vk.Vm+
v.~4v+ V, NV~-hw:
23) \\ ~./ J 29)
whe~e th con t mts of 6he model, for convemence
24) mbolicclly dffmted by ¢~, me obtamed from
¢,=F,f,+(I—F,)~
7he ¢1 set of constmtsis6he one ofthe t mdardk—m
model md the ,6 set of constmts has beff~ deriv d
fiomthek—smodel mdisgiv nby:
~ = 0 4404 i h~ = 0 09 i h: = 0.0828 j
(t >~=1 j (qw)~=1,l7 j
7he blendmg function, F. is giv by
F. = t mh (org4) j
(3o)
wi6h
I ~ ~ SOOv 4(qw)2k
t/rg=mm LmaX ~o 090)Yri y 0) i CD~
26) (31)
md
CDKW=maX(2(q~20~3 t~ ~ jlO-~° (32)
k VS+V ((V+Vt)Vk)—hS k 27) 2.35 q—S model
md
Dt S+V ((v+` )Vm)—hm . 28)
7he m equationc mbe integ cted d wnto the wall
md the model con t mts me:
c = 0.5532 j h~ = 0.09 i h = 0 075 i
t =2 j q~=2.
2.3.4 Menterts k—m model
M nter's v rsion of 6he k—m model, [13], is c
blendmgbetw enthek—~ mddhek—mmodels 7he
objectiv of 6he model is to solv the m equation in 6he
near wall ~egion, which does not ~equi e exhc damp-
mg f mctions, wherecs m 6he octer region of the flow
6he ~ h m port equation is solv d
As m the t mdard k—m model,6he eddy viscosity
is computed from 26) md the k t mspo t equati m is
7he q—5 model is proposed m [14] It is c two-
equation model deriv d from 6he k—~ m odel wi6h the
objectiv of having tmbulent q mtities which go to
vmo et the wall 7he two turbulent q mtities of the
model, q md 5 are rehted to k md ~ by
(33)
7he h msport equations of q md 5 are deriv dfr m
6he h mspo t equations of k md ~ with the rehtions:
Dq = I Dk
Z)F 2qNf
D ~Ds _ ~Dq
Ot ~ ~T q Ot
~ 6he p~esent impl menbtion of the method,
hien's low Rey olds v rsion of 6he k—~ model was
cdopted to obbin 6he h msport equations of q md 5
7he eddy-viscosity is giv nby
v~ =C~f~25
(34)
(3s)
ad
7 he near wall dampmg fumctions are giv n by:
D] vtS + V . ( (v + Vt ) Vq)—S—y j (3 6)
D; C ~vS C ~a C 5
~F Ciq~- <2f<2q- 43f<3~+
V ((V+ t ) VS)
(37)
f~ = 1—e~°-°l 15Y~ (38)
C<:f<: = 2.6 - 0.792e (~) j (39)
1~N
7 he model const mts are
C<3f<3 =2e ~ J_I. (40)
cp= 0.09 j C
6he skin friction coefficient for the flow on z flzt phte
Ther for, only 6he fl st option is lef to zvoid the use
of z g id-dependent bo mdary c mdition in this paper
w shall compme two zlternativ s: i) zdopt equation
(43) with N~ in hding viscous conections to deflme
6he n ar~ll values of m, BCI; md ii) obtzm m zt 6he
wall fi om equation (44), BC2
3 Results and Discussion
3.1 General
All calcuhtions w re carried out with the computer
code PARNASSOS, [19], which solv s the R y olds
Av raged Nzvier-Stokes equations m thei complete
form [20] The test case m 6his paper is the flow
aro md 6he D e ~dy te y) tmker which has ear-
lier been subject of c mparativ computations m 6he
Gothenburg md Tokyo Workshops, [21] md [22]
Two main reasons justify this choice: the flow is
sufflciently complex to test 6he ccuray of 6he tm-
bulen models, md, in particular, it e hibits, zt lez t
zt model scale R y olds m mber, the socalled 'hook
shape' of the isolmes of the axial v locity zt 6he end
of 6he stern, zs z r sult of the existen of shongbilge
v rtices A detailed m merical v rff cation st dy has
ben pe formed with PARNASSOS for this test case,
bodh zt model scale R y olds n mber, [3], md full
cale R y olds m mber, [4], which permits 6he selec-
tion of z g id with sufflcient resolution
~ 6he pr sent tudy, flv dffferent Rey olds m m-
bers hav been consider d: 5 x 1o6, 2 x 107, 1Os,
5 x 108 md 2 x 109, with 6he R y olds n mber d -
flmedby
R =—.
v
A Cartesi m coordinate system is inh oduced with
6he z axis zlong the mdisturbed sheam, the z axis
v rtical positiv pomtmg upwards md y completing
z right-h md system The origin of the coordinate sys-
tem is located on the forward p rpendicular zt 6he ship
mmet y phne on the k cl Ime All the variables pr -
sented cc made non-dimensional using U~ md l zs
6he v locity md length r fer n e cales
The c mputational domain covers only 6he flow
fleld n ar 6he tem The inlet md outlet phne are z
c mst mt pkmes The inlet pkme is located zt z = 0.5l
md 6he outlet phne zt z = I.25l Th extemal b md-
aryis mellipticalcylinder,giv nby:
y ~ /z—0.056lN
(0,149) +1~ 0,140 J =
The r mainmg bo mdaries are 6he fiee surice, pkme
z = 0.056l, 6he symmeby phne of the ship, y = 0,
md the hull surface
The vol me g ids w r cr zted wi6h z proprietary
elliptic PDE g id gen ztor, based on 6he GRAPE zp-
proach [23] The m mber of g id nodes in 6he sheam-
wise md gi thwise di ection is the same fcr 6he fl e
R y olds m mbers: N; = 161 md N: = 41 The
n mber of g id nodes in 6he normal di ection, N~, m-
creases wifh R N~ = 81 for fhe low st R md 10
g id lin s cc zdded each time R is mcreased, which
leads to N~ = 121 for R = 2 x 109 The g id lin
p cing in the ncrmal di ection is deflmed by on -
dim nsiorul str tching f mctions, which are t med to
obtamamaxim mvalueofy ztfheflsthyerofgid
nod s zway fiom fhe ship smfce of zpproximately
05
Fiv sigmiflc mt flow p~meters w re selected to
compme the dffferent m merical sol tiom:
e Friction r sistance coeflflcie t~ CDI:
CD!
(~3U') 14 xo<|did
-pU l:
e Pr ssure resi tance coefflcient, CDP:
CD =2Jcp(4xo<) ~did5
e Wzke frction, Wf:
W = 1: (I— ) d
e Maxim m cross-sheam v locity zt z = 0.989l,
(Vw~m~,, with Vw = J:u )2 + (U3~.
e Minim m axial v locity compon nt m fhe flow
fleld. Um~r
The maxim m cross-sheam v locity zt z = 0.989l
is r hted to fhe bilge vortex intensity md U~ ~r identi-
fles fhe exi ten e of sheamwise flow separation The
mteg zls in hded in the deflmitions of CDI. CDF md
Wf are evaluated with G mssim quadratme rules as-
summg zbi-lmear variation of fhe urJmownsbetw en
fhe g id nodes The arez Q for the calcoktion of the
wake fi ction is fhe propeller di c, which has be n IO-
cated zt z = 0.989l wifh fhe axis of the propeller zt
z = 0.0166l; the die radius is R = 0.015l, while z
ero hub mdius h~s been zss med.
3.2 Wall Boundary Condition for tO
To mv stigate fhe i flu n e of fhe m mericel im-
plementation of the wall boumdary condition of P.
Varietle
CD! X 1 O
CD X 1 O
Wf
uW)m:~
mir 4
L.Vt~me' X 10
(V~>m~; X 1O
BCt
1 944
0911
0 609
0 343
-0 075
0 265
I 582
BC2
1 612
0 878
0 565
0 365
-0 076
0 257
I 504
Tetle 1: Comparison of solutioms obtamed with
Mbuter's k—P model usi g differ nt implementa-
tions of the P wall boumdary condition
w hav calcuiated the flpw et f = Sxl06 with
Mbuter's v rsion of the k—P model with fhe two op-
tions considered: BCt, which obteins P from fhe th -
or ticel valu for y < 2.5 md BC2, which is based
on m od hoc deflmition of c flmite valu et fhe wall
Tetle I presents the fl e selected flpw qumtities md
fhe me m md maximum valu s of v~ obtamed with
BCt md BC2
The dffferen s obtain dbetwe n fhe two solutions
are certamly not neglig~ble As on might expect, fhe
fi iction resist mce coefflcient, CDf, e hibits the largest
d fference How v r, the limitmg treamlin s of both
prlcuiations, which are depicted m flgme I, are simi-
~ BCZ, (T~)c
Figme 1: Limiti g sheamlmes for Mbuter's k—P
model wifh dffferent wall boumd uy conditions
At fhe propeller pime, x = 0.989l, fhere me sig-
nffc mt differen es between fhe isolin s of the axiel
v locity, as shown in flgme 2 With BCt the speed is
deflnitely Ipwer in the i mer wake th m with BC2
0.0<
\ 0.01
A rlC
0~00 0 0 i O.O2 0.03 0 04
02
os
04
05
~ o:
~ 0.7
v o:
~ Oq
Y/~
Figme 2: A ial v locity isolin s et x = 0.989l for
Mbuter's k—P model wifh d fferent wall bp mdary
conditions
These r suits show that th flpw prediction is
clearly dependent onthe m mericel implementation of
fhe P wall boumdary condition They suggest ft et the
P behaviou et the wall c mhardlybe seen es c shong
pomt of the k—P models From the r suits it is not
clear which is fhe best choip,BCt or BC2, how v r,
as discussed etov, the re mlts of BC2 me irJ~erently
g id-dependent Therefore, w will edopt BCt for the
r meinmg prlcuiations wifh fhe k—P models
3.3 Scaling EEects
The results of fhe fl e selected flow qumtities md
fhe maximum valu of v~ me giv n m table 2 for the
fl e R y olds numbers md for th various tu bulence
models te ted ~ st mds for the maximum dffferen
betw en th predLctions of th different tubulenee
models et c giv n Rey olds m mber Th dffferen s
betw en predictions wifh dfferent tubulen e mod-
els come out es mprecietle et model scele R y olds
number but tend to diminish with the mcrease of the
R y olds m mber A e peption is foumd in fhe maxi-
mum cross-sh eam v locity et the propeller pi me; it is
fhe only on of the selected flow varietles which does
not ch mge m on tonicelly with the Rey olds m mber
Figme 3 pr sents the friction resi tanee coefflcients
giv n in tdole 2 We hav tried to compme the r suits
wifh two fi ietion Imes, the ITTC lin .
(CDf)° {i 'f ~ 2)2 i
Va~ nble
CD!
xlo4
Cl o4
Wf
(Vw~mcD
Ul ~r
(V~)mm:
xlo4
SXIO6
2xl07
108
SxlO~
2xlO9
sxlo6
2xl07
108
SxlO~
2xlO9
SXIO6
2xl07
108
SxlO~
2xlO9
SXIO6
2xl07
108
SxlO~
2xlO9
SXIO6
2xl07
108
SxlO~
2xlO9
SXIO6
2xl07
108
SxlO~
2xlO9
CS
I 573
1269
1 22
0846
0732
0598
0527
0466
0423
04 2
0528
0462
0397
0347
0315
0209
0199
0203
0206
0208
-0012
0004
0 000
0 000
O 000
2 327
1 979
1 647
1 374
1173
Sf
1 816
1 469
1177
0965
0825
0788
0720
0649
0605
0578
0632
0547
0452
0393
0352
0280
0301
0302
0294
0263
-0059
-O 055
-0035
-O 005
0 000
1 627
1 338
I 093
0917
0780
MT
1 606
1 335
1 098
0916
0791
0741
0648
0583
0547
0535
0619
0535
0454
0393
0353
0290
0289
0276
0252
0228
-0068
0040
-O 001
0 000
0 000
1259
1 026
0845
0711
0626
KE-TL
I 561
1280
I 059
0885
0767 ,
o7l3
0638
0580
o s43
0542
0563
0493
0421
0369 1
0334 1
0287
0286
0262
0229
0227
-0028
-0007 1
o ooo 1
o ooo 1
o ooo 1
lil3 1
0927 1
0798
0660
0556
KE
598
3ls
082
9oo
0776
0687
0645
0604
7o
0565
0576
0512
437
0380
0341
0266
0284
0284
0256
1 0229
1 -o ols
-0016
1
1 o ooo
1 o ooo
9s
0965
0823
0 708
0586
rQZ I
44
ll97
0988
0824
o7l3
o 655
0614
o 589
os7
o s7
o 532
o 479
o4
0 365
0 330
0 263
0 271
0 268
0244
0 228
7 1
1
o ooo 1
o ooo 1
o ooo 1
1 046
o 889
0742
0 603 1
osos 1
KW
1 939
I 546
1 213
0 981
0 841,
o 923
o 829
0744
0690
0 648
0616
o 532
o4
0384
0 345
0 347
0 378
0 396
0 395
0 336
7
065
4
27 1
o ooo 1
492
l 318
o 93
0 763
0 700
K\AfM
944
47
1221
983
0 843,
9
0817
0732
0675
0638
0609
3
449
0386
0345
0343
0367
0377
0369
0331
7
-0 064
-°°41
1 -0 034 1
1 oooo 1
582
1361
4
0862
0756
r~
499
3so
0233
4l
3o
325
0 320
0278
0267
0236
4
085
7
0 046
0 038
0 138
0 179
0 193
0 189
0 128
063
061
4l
1 o o34
1 o ooo
281
9o
9os
800
0 668
Table 2: Comparison of solutions obtamed with the
md the Schoe herr line,
0 242 = loglO (Rn(CDf)O~s) j
Smce our computation domam covers the afi haff of
6he hull only, w have estimated 6he equivalent plate
fi iction of 6he aftbody as
Di)pl [2(CDI>O(.R )—(CD )O ( )~ T
whe~e Sw is th wetted surface of the ship included
m 6he computatiom~l domam, which is ass med to be
half of the total w tted surface The p~edictions of a11
6he turbulence models e h~bit the correct hend with
sewval turbulence models at different Rey olds n mbers
6he mcrease of the Rey olds m mber, but there is
clear diffe~ence m slope
As ~ fur6her relevmt result, w have plotted the
ae m wake fracti m, Wf, as afunction of the Rey olds
~ mber m figme 4 It is interesting to note 6~t m both
igmes 3 md 4 the~e is good ag eement betweff~ the
iA model md the two k—m models, K\Af md K\AfM
The calculated limitmg sheamlines at R = 5 x I o6,
b'= 108 mdRn=2 x lO9are illu trated mfigmes 5
o 7 fcr th models CS, ~iA, MT, K\Af, KWM md KE
As in the previous re mits, 6he dffferences betw en
he predictions of the various tmbulence models tend
o diminish with 6he inmease of Rn Once mme, the
2.5F
2.0
o
~ ~ 5
. . . _
1.0
0.6
6.
~ MT
; td
W?
:. ttt
ITTC \<
Schoenher:
+CS *SA hKE
O MT ~ KE TL
f QZ
1~ KW
0 KWM
7 6 9 10.
LogtO(Rn)
Figme 3: Friction resistmce coeffcient of the cft-
body, CD! CS C f motion of the Rey olds m mber for
fhe variocs turtclence models tested
0.65F
o 6ot
o 55t
- o 50t
o 45t
o 40t
o 35t
O 30~
6
+C3 *3A xKE
* OMT ~ KE TL
~QW
x O KWM
A ~
+
k3
+
3
+ ~
+
7 6 9 10
Log t o (Rn)
Figme 4: Wcke ficotion, Wf, es c f motion of fhe
Rey olds m mber for the seve~al turtclff~qe models
tested
~esclts of the SA, mw md mw v~ models are very sim-
ilar: At full qcle, f = 2 x 109, the CS, MT md KE
also show good cg e ment
he axicl velocity isolines et fhe propeller pkme,
x = 0.989l, et fhe same f ee Rey olds m mbers,
5 x 106, 108 md2 x 109, are presented infigmes 8 to
I O he turbulence models i qlcded are cgain the CS,
SA, MT, KW, K\AfM md KE he Ul isolines e hlbit
Figme S: Limitmg sheamlmes ctf = S x 106
c d c tic i fiue qe of f At model scale, fhe t pical
hOOk shape' does cppear for th k—m models, md
to some extent, for fhe one-eq wtion mod 1s5 SA md
MT Hqwever, et x = 0.989l, the 'hook shape' tends
to discpp ar wifh fhe i q~ecse of fhe Rey olds m m-
ber At f = 2 x 109, nqDe of fhe p~edictioms e h~bits
c hOOk shape' md d fferff~qes betweff~ the resclts of
fhe variocs m odels, i qhding the clgebrcic CS m odel,
are rcther smell
his effect of f is ~eMted to the st~etch ng of the
b ilge qrt :x, genemte d wif hin fhe ship b o mdary Icyer
which red ces its thick ess with the i q~ecse of f
Figmes I I to 13 illc trcte the cross- trem velocity
field et x = 0.989l for fhe same f ee Rey olds m m-
bers he plots i qlcde fhe CS, MT md K\AfM mod-
els Although there me some dffferences betwen the
p~edictions of the f ee models ew~n et f = 2 x 109,
xsm co e Deo ,mese remoco om eeas~ylmprov
wi h s s mple ooneotlm t the grodcotloc temm o the tr m
eqc diOC of he twbulect qc mtity
Figme6:Limiting treamlinesatR =108
The tretchmg of The bilge vo tex with The increase of
R is clear for the th ee models
We should note 6~t The present results do not im-
ply 6~t the t pick hook shape' of The al velocity
isoh es disappears with The inverse of R. it just m-
pearsfmtherdow sheam Figmel4presentsth ve-
locity field et x = I.ll obtained with the IC.'JM model
for Rn = 2 x 109 At this location, the bilge vo tex is
almo t axisymmetric md The Ul plot shows The typi-
cal 'hook shape', which is mder t mdbly w cker 6 m
et the propeller plane et m odel sac e, bec mse The bilge
vortex has not only rolled up m the near wake but it
has also diffused
4 Conclusions
We have presented remits of c m meri cl inves-
tigation of scaling effects in ship stern flows using
algebraic , one -e quct ion md two ~ qua t ion turbulence
Figure 7: Limiting mecml es et R = 2 x 109
models The turbulence models w He all implemented
without my special t ming d pendent on the R y olds
n mber
For the two-equation k—m models, we have
pouted out the deficiencies of c widely accepted mm-
mericcl implementation of The wall bo mdary condi-
tion of m
The results of The ccicoktion of The -I w aro md
The IA e (mystery) t taker et -i e R y olds m mbers,
5x 109,2 x 107,109, 5x 109 md2x 109, suggestthe
following conclusions:
e it is possible to simulate m merically ship stern
-I wsfrommodeluptof llsccleRey oldsmm-
bers with the most popular eddy-viscosity tmbu-
lence models, mcludmg algebraic, oneequation
md two-equation models
e in global temms, The predictions e habit the same
Tend in the flow field with The increase of the
Rey olds m mber for all the turbulence models
A AA
A na
A A
0.0
A A
U.U ~U
0.01
0 00 0 01 0 02 0 03 0.04
~.u
A Aq
A A
A A 1
y/L
A A
U.U ~ UA
MT
0.01
0 00 0 01 0 02 0 03 0.04
.u
A Aq
A A
A A 1
A A
t 01
* OZ
O Os
04
as
~ as
s 07
os
os
y/L
0.0
0.03
0.02
0.01
0.00
0.00 0.01 0.02 0.03 0.04
y/L
0.03
0.02
0.01
0.00
0 00 0.01 0.02 0.03 0.04
y/L
0.01
0 00 0 01 0 02 0 03 0.04
y/L
Fit3~re (i: Axi4i vebc~t isolines 4t ~ = 0.989l ob-
+ ot
* 02
0 Os
04
as
~ cs
07
08
Og
+ ot
* 02
0 Os
04
cs
~ cs
07
08
~ OD
0.03
0.02
0.01
0.00
+ ct
* Oz
0 o3
o4
os
os
~ 07
os
~ OD
000 001 002 003 0. 04
y/L
Fit3~re 7: Axi4i velocit isolines 4t ~ = 0.989l ob-
Do
N
0.0
0.0
0.0
A n
r/
w.u (S
0.0
0 00 0 01 0 02 0 03 0.04
y/L
0.04 ~
0.0
0.0
N
.1
~ I
0.0
A ~
oo h
oo h
1
0.01
A ~
y/L
0.04 ~ ~
w.w
KE
0.01
0 00 0 01 0 02 0 03 0.04
y/L
Ot
OF
04
04
o~
06
07
09
09
Fit3~re 8: Axi41 velocit isolines 4t ~ = 0.989l ob-
hined 4t Rn = 2 x 109
~ ~A
... .
0.01 g~) ~
0.00 0 01 0.02 0 03 0.04
N
0.0
0.0
y/L MT
·~W
0.00 0 01 0.02 0 03 0.04
y/L KWM
i>3~re 9: T~nsverse velocit field st ~ = 0.989l ob-
hined 4t Rn = s x I o6
o.ol
o.oo .
0.01
M
O.01 ; V
0.00 0.01 0.02 0.03 0.04
y/L CS
004
0.01
0 00 0 01 0.02 0 03 0.04
y/L MT
004
0.01
0 00 0 01 0.02 0 03 0.04
y/L KWM
Figure 10: Transverse velocity field et ~ = 11.989l ob-
tained et Rn = I 09
~ i
\
N
\
N
U.U
0.0
~ fit
0.0
0.0
::~ ~
0.00 0.01 0.02 0.03 0.04
y/L CS
0.00 0.01 0.02 0.03 0.04
y/L MT
AAt i ;~\
o.ol Alto ~~'~Ir~I~;~ ~~ v \ ~~ N ~ ~ \
0.00 0 01 0.02 0 03 0.04
y/E KWM
~ ~t3~e 11: Trarisverse velocity field et ~ = 0.989l ob-
tained et Rn = 2 x 109
\
~ . ~ ~
nn~
nn~
nn
not
nn~
002
nn,
n nn
0.01
ooz
0 oo 0.01 0 oz 0 03 0 04
+ 07t
~ 076
0 o79
~z
09s
ggs
~ 09t
~ 094
y/L
Figme 12 Velo ity field et I / l bbm d with
Menter k t m del et Rn 2 x ;O9 Above axwl
velocity, U', isolmes Below: cross- tream velocity
field
e The dismep mcies betw en figw fields obtamed
with diffe~ent tmbule ge models et c given
Rey olds n mber tend to decr cse wifh the m-
crecse of the Rey olds m mber
e Although fhe perfommance of the k—t models
seems to be wxy e souraging. fhe predictions
depend on fhe m merical implementation of fhe
t bo mdary conditigm et c solid surfae Some
implementations cdvised m fhe open literatme
are mass ptable
The present ~esults ~ei force fhe need for relictle
experimenbl dat4 et full secle Rey olds n mber for
41idation p gposes All eddywiscosity turbulff~se
models, used here, w re essentially developed for
bo mdary-kyers et modemte R y olds n mbers
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