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OCR for page 910
Ship Stern Flow Calculations on
Overlapping Composite Grids
B. Regnstrom, L. Broberg, L. Lar iSOII'
( FLOW ECH nternational AD, Chalmers University of Technology, Sweden)
ABSTRACT
A method for predicting the viscous flow
aro md ship terns is presented its main cdv mtage is
the flexible high-quality g id on which She go coning
equations Ed the bo mdary conditions are dircreti:D:d
A set of overlapping g ids on the hull surfae are cre-
cted either by hyperbolic marchmg from one of the
bo mdaries or by cutting the surfae by horizontal Ed
vertical plumes Body-ftted vol me g ids are then
g own hyperbolically out from the surfae At the out-
ermo t edge of the compnhtioncl domain c bak-
g o Ed Cartesian g id is chosen Ed c sequence of finer
Ed finer Cartesi m g ids is automatically generated to
create c s fficiently mooth t msition betw en the
coarse edge g id Ed the tine body-fitted g id The
algorithm guar mtees duct Here is s fhcient overlap
betw en all g ids
The Rey olds-Avemged Navier-Stokes equa-
tions are solved on the overleaping g id using finite
difference discetization The equations are partially
tr m formed Ind all variables are co-located Pressure
Ed velocities are coupled vie c 5 ah L'L E clgorif m Ed
Rhie Chow mte polation is used to avoid checker-
board oscillations Computed results are compared
with measured data for th ee different hulls
INTRODUCTION
The state of She art of Computational Fluid
Dynamics applied to ship desig was review d et the
previous Symposi m on Naval Hydkodynamics by
Larsson et cl (1998) While She obtainable CFD acu-
ray is s fficient for m my purposes, erpff icily when
optimizing the hull shape, q mtitative predictions of
m my hydkodynamic q mtities must still be regarded
with caution Several reasons for She lack of absolute
acuray were listed Ed discussed in She review
Examples are inadequate g idding, dissipative m meri-
ccl tffhmiquer Ed too approximate turbulence mode-
ling Free surfae representation was also mentioned es
m area w re further developments are needed
Dming She pest five years effo ts have been
mad in She research g oup et Chakmers LOWTECH
to improve the state of She art m all four areas The
work on g id generation has been reported in Petersson
(1997c,b, 1998) ad Li (1998c, b), while m mericcl
developments have her presented by Carlsson (1997,
2000) Ed Carlsson Ed Petersson (1999) Turbulence
m odeling was reported by Svemmberg (1997, 2000) Ed
Svemmberg et cl (1998) Ed f ee surfae developments
by K mg (1996,1997), Vogt (1997, 1998), Vogt Ed
K mg (1997) Ed V Oft Ed Larsson (1999) ~ the
present pCpff m improved g iddmg technique is pre-
sented together with She newly developed sol em, celled
CHAPMAN Free surfae developments md cdvarmed
turbulence modeling are not addressed The g id gener-
ction is presented m She next section, which is followed
by c section on the Na vier-Stokes solver Thereafter the
validation of She medhod for th ee different hulls is pre-
sented md finally some comments md recommendc-
tions for f tore work are given
GRID GENERATION
Most ship flow calcoktions presented mtil now
have exploited She fat that c ship hull is c mooth
bo mdary, mlike t pick bo mdaries of, for in tarme,
internal flows in fat in the vast majority of cclculc-
tions to date single block shuctmed g ids have been
used (se Larsson et cl (1998)) C rtain cdvmtages c m
be envisaged, how ver, m using more cdvarmed g id-
ding tffhmiquer even for 6 is relatively simple geome-
try Thus, more complex regions, such es the stempost
md the possible stem bulb, m ight be better resolved A
high resolution in These regions is importmt for the
prediction of She flow m She propeller plan just
behind Father, the singmbrity line(s) m She g id in
front of md behind the hull m be avoided A other
cdvmtage is that the flow near She free surfae can be
computed in a Shin g id with sufficient resolution
If appendages are added, She geomet y md pos-
sibly also the topology of the domain becomes more
complex The appendages Themselves are how ver
OCR for page 911
mostly sheamlined There are thus c limited n mber of
flow regions, each of which having sm oodh bo mdaries
A completely mm ;chned g id in this case would
waste the regmbrity offered by the bo mdaries it
should be better to take cdh-mtsge of the smoothness in
ecch region md to create separate shuctmed g ids in
ecch such domcm The que tion is how to join She dff-
ferentg ids Buttjoinedg ids are usually urmecessarily
restrictive A more flexible solution is to let the g ids
overlap arbitrarily md to handle She overlapping parts
in c separate procedure Component g ids with suitable
resolution of the bo mdary Icyff c m the be fitted to
ecch part of She hull md Cppff dages, while She outer
flow is computed in c Cartesim or cylindkiccl g id
O ffiappmg g ids have been m use for clout
ten years md the most w 11 I:noss n code for generating
such g ids is CHIMERA, originally developed et
NASA in She USA this g iddmg technique is of en
referred to es the CH MESA tech iqw A special
series of symposia is devoted to this g id type md the
5th Symposi m on Overset Composite Grid & Solution
Techmology will be held et She University of Cclifomic
et Davis, USA in September 2000 Applications in
hydkodynamics have been presented by '.\ ems et cl
(1994), Lin et cl (1998) md Mcs ko (1998) it should
be pouted out that other cdvarmed g idding techniques
are also rapidly bemg mtrodued m hydkodynam ic cal-
culations Most popular is She stmdard multiblock
technique, see e g Beddhu et cl (1998) md Wilson et cl
(1998), but mstructured g ids have also been used, e g
Hmo (1998) md Lohmer md Ovate (1998)
The present grid generator
The g id ge fiction process starts by importing
c CAD surface description file Often These tiles have
to be improved, for instmce by closing the propeller
shaft opening et the stern A separate module creates m
ellipsoid that is fitted to the bossing md faired to the
hull Having fi ed She surface, surface g ids are get r-
cted on She hull ( md the Cppff dages, ff my) This c m
be done m one of two ways The simplest possibility is
to cut the surface by horizontal or vertical plumes to
obtain one set of lines The other set is obtained thffe-
cftff by com cting pomts et c given percentage of the
total arc lend h along ecch one of these Imes, measured
from one of the patch bo mdaries ff g ids of f is type
get too skewed hyperbolic marching is used Starting
from c patch side, g ids are g ow inwards in c step-
wise maimer One step consists of moving all he pomts
on one Ime to c new one This is done for ecch pomt by
taking c rep et right males to She original line md in c
direction t mgenticl to She surface The pomt fo md will
generally be away from the surface, so it is moved
along the normal doss to the surface Each step length
is detemmmed f om the cell area d-fined bv She tart md
end positions of two successive points The g id may
be forced to follow given lines et She side bo mdaries
Having completed the g id generation on the
hull, body-fitted vol me g ids are g ow hyperboli-
cclly outwards from She surface The procedure is c
th e - dimens ional generclizat i on of the two - dim en-
siomd one just described Points are fir t moved et right
males to She hull surface md ah reader et right males
to She surface defined by the g id points from She previ-
ous step The silos of the step, md She capability of fol-
lowing side bo mdaries are es described groove
The description so far concerned the curvilmear
body-fitted g ids These are embedded into one or more
backg o md g ids, which are normally Cartesi m (cylin-
dkical g ids have also been used) md extend to the
bo mdaries of She computational domain
When cil component g ids have been generated
the o Up algorif m starts Each g id is given a
mique priority, She initial backg o md g id always
being the loss t The overleaping g id is then con-
structed m the followmg steps:
I Cnt surface holes, i e mark all points outside of
the computational domain md Iymg on g id faces
that are part of She physical bo mdary as dead
2 Cnt vol me holes, i e mark She remaining pomts
that are outside of She computational region as
dead
3 Set up exact mtemolation points
4 Ckssffy (i itchy) all remaining pomts, see below
5 Check the consistency of She interpolation points
6 if there are my bad mtemolation points, refine the
backg o md g id md go to I
7 Trim unneeded interpolation points
Except for rep 6, 6 is is the same algorithm as
Peterson (1997a,b, 1998) After step I md 2 all pomts
that are outside of She computatiorud region are marked
as dead The exact interpolation points m step 3 occur
where a g id has been split in order to acommodate a
chmge of bo meaty condition The g ids share the
same mapping f motion md overlap by one g id cell
md consequently two layers of g id points coincide
The impo tance of 6 is rep is that She interpolation
stencil for These pomts has zero width so there is no
risk for it to incorporate my dead points in the mitial
classification the following is done for each unclassi-
fied pomt in each g id, starting with the highest prior-
ity:
· Check if it mtemolates from a higher priority g id
· if not, check d' it c m be a dearth on point
· if not, check if it mterpoktes from a I wer priority
grid
· if not, mark point as dead
OCR for page 912
Af er step 4 ecch g id point is ckssffied es m
intffpohtion point, discretization pomt or deed pomt
Pomts in nonbackg o md g ids md not close to physi-
ccl bo mdaries are clways cllow d to intffpohte to the
backg o md g id md refmements thereof ff 6he back-
g o md g id is till too coarse to yield m intffpohtion
stencil free of deed pomts, the mteqpohtion pomt is
marked es bad
In step 5 cll interp o k t ion md discretiza ti on
stencils are in pected md ecch pomt that uses c stencil
that conbins c deed pomt is put on 6he list of bad
points All points m the stencils of bad intffpohtion
points due to too coarse backg o md, see ctov, are
clso put on this li t if the list is not ffmpty after cll g id
points cre chssified, the bad points m 6he fine t back-
g o md g ids cre used to cclcokte c set of boxes 6~t
encloses the bad points, Bell et al (1994) For ecch
such box c refmed Cartesim backg o md g id is
inse ted in 6he ovffhppmg g id, md the clgori6 m re-
starts f om 6he begi ming
S OLVER
Govffrdug equations
In c Cartesi m coordincte sy tem 6he Rey olds-
averaged Ncvier-Stokes equations for mcompress~ble
flow may be written
OUl + Uj aa: j' + a a (vaa: j') aaiJ = 0(1)
md the contimmity equation reeds
aul
, =0
oi l
Us cre the me m v locity components md Xs the
space coordinctes P is 6he pressure, v the kmematic
viscosity, ~j 6he R y olds shess tensor, md t is the
time
Cartesi m coordim~tes may be used m 6he back-
g o md g ids, while the body fitted g ids require cmvi-
linear coordinctes Tr m forming only the mdependent
varibles 6he ctov equationsbecome
md
a + dt ut + d
I ( jt a ) La
(3)
(Jd U ) = 0
(4)
In the cclcuhtions of 6he present pcper 6he two-
hyer k-c model of Chen md Pctel (1988) is used, i e
tr msport equations for k md ~ are solv d m the major
part of the compubtiork~l region, while c prescribed
lengh scale is used together wi6h 6he k~quation m c
thm hyer close to the wall The h mspo t equations
may be w itten
at+Ulai`=a ((V+~jt)ai`)+
md
a +ula =
a ((V + ~ ) ) + j-(C~ ,Pt C~: ~ )
(6)
These m be h msformed to curvilinear coordi-
m~tes like whff (1) is t msfommed to (3) The produc-
tion term is
au
Pt =Tva
(7)
with ~ j cclcokted using 6he Boussmesq cssumption es
follows:
(2) ~`j = v:Sv (8)
s aUl + a
lj a a
(9)
This m ms that 6he kst term m (1), m be h m-
dled by cddmg the turbulent viscosity v: to the kine-
matic viscosity
v: = C~—
Theconstmtshav thei traditiom~lvalues:
(10)
C~ = 0 09 C~ I = 1 44
cyt = 1 0 C~: = 1 92 (11)
o~ = 13
OCR for page 913
The k equation (5) smd the prod3ction temm (7)
9 e ms mts med m 6he rs4s -ws 11 kyer, b 3t the r equation
(6) is red sed to:
3 = 1
smd the calc'
ch mged to:
Ic3ktion of 6he eddy vi sosity (10
v, = C~/~lk
, withy6he normal distasw4 to 6he wsll:
Ry = Rey p
cl = ~C 3/4
/~ = cly(l exp( A ))
/~ = cly(l exp( AY))
The im ff kyer extends f om the surfs s to
Ry = 250
Numeneal method
A~ = 70 A~ = 2c
= 0 418
In order to mske the intffpolstion equations 9 s
simple 9 s poss~ble smd to keep 6he m mber of mterpok -
tion pomts mall some cs e hs4 to be exe six4d when
discreti ing 6he equations Fir t 6he discretization sten-
cilhs4 tobe ss small s4 possible, othe wise the ov4rlap
regions will be wide Si se 6he equations 9~s second
order the ms 11est ste sil possible is 3 nodes wide Sec-
ond, collocsted storage is req3ired smce taggered stor-
sge givss four mtemolstion pomts per cell Collocsted
stora ge s Iso fs silitstes the 3se of Cs tesi m compors4nts
so 6~t base v4ctors will not hav4 to be t msfommed
when interpoktmg betwsen the g ids Th3s, some
medhods I ke higher order 3pwmd differff ses smd stag-
gered g ids cs mot be 3sed, evsn tho3gh they 9 e
mmerically sthastiv4 The challenge in the present
work is to fud 9 ststle, efficient smd 9 surste sheme
with collocated node-cenhed storage wi6h 9 th ee node
wide ste sil
Anodher requested property of 6he medhod is the
capability to sim 31ste thm bo mds y Isyers At model
scale the Rey olds n mber for 9 ship h311 is t pically
107 smd st f 311 scs le 109 To resolv4 the corresponding
bo mds y Isyffs, g ids with cell 9 spect rstios 3p to 106,
Reg trom (1994), E 9 smd Hoekstra (1996), close to
the ws 11 sse rs4eded if 9 time- teppmg scheme is 3sed
it m 3st not hav4 too sevsre limitstions on the time step
sin4 for s 3ch g ids smd this e 91 3des purely explicit
schemes For implicit smd semi-implicit methods the
high 9 pect rstio cells will givs 9 poorly conditiors4d
coefhcient mstri if 9 hs ditiorud shetched g id is 3sed
(12) sm iterstiv4 method capable of h mdlmg this m 3st be
3sed
) is The kst req3i ement on the slgorithm is 6~t it
m 3st be rob 3st Evsn ff this m ms lowsr 9 scura sy the
re s 31t s cam be 3x4 d 9 s 9 bs s is for im prov sm ents The
13 discretization of the govsrnmg equations smd the
( ) bo mds y conditions will now be de sr~bed
A shif operstor is defmed by:
Enf(~,, .~j. '~N) f(~,. .~i+n. ,~N) (15)
Normally 6he mdex i will refer to ors4 of the
coordincte di ections Using the opffstor the second
(14) order cenbal smd fi st order fo ws d smd bsckws d
finite dfffere sess edefrs4dby:
E' E'
i 2s
3+s Es
s
(16)
(17)
(18)
Next the temms in the ps tislly hamsformed
equations 9 e expressed by repls smg the ps tis I deriv9 -
tivss wi6h fmite dfffere ses This is dors4 3smg the
operstorsdefrs4dmT~olel:
OCR for page 914
p:
pemtol
Convection (Ist
mder upwind)
Convection (2 d
mdel cenhal)
Dfff sion
kadient
Div Igence (fcce
cenhed)
Div Igence
Lcpkci m
Symbol
K
lu
~c
Tctle 1: D fmition of operctols
_ Ca~tesi m fmm
max(O,vj) jl+min(O,vj):l
_ Cmvilmea~ fmm
max(O,vj) jl + min(O,vj):
_ Jj
J i (Jgj v~ il/~)
_ j
i j/~(v~ j/~)
h ( i) i
o~ij
1 1/2
h(i) ij
1 1
h(i) ij
(h ( O ) J. ( j )
ij(Joj)
il/~(Jg
whele fhe conha~i mt v locity components a~e
Vi = ui/h(i) in Ca~tesi m md vi = ojuj in cmvilin-
ea~ comdinctes
The SlAdPLE scheme is used wifh fhe Rhie-
Chow (1983) method to suppless chequer boa~d oscil-
lations that will of helwise occm when usmg collocated
stmcge Fi st comes c predictm step that is implicit in
u:
u +AtKu AtVu = u AtGp (19)
Superscript wifh one m mme ~ ks m
cpproximation to fhe va~i~ole et time lev I n+l The
operctol K is chybrid of fhe fn t md second mder con-
v ction operctols:
K= clK:c+(l Ct)Klu
(20)
The cmlector step is explicit m u md implicit in
u +AtKu AtVu = u AtGp (21)
The diffelence betw en fhese is
u = u AtG(p p )
(22)
Tcking fhe dismete div Igence D of this equa-
tion results m c spa~se Lcplacim DG that will giv
solutions wifh chequer boa~d oscilktions its tencil is
clso widel f m fhe th ee nodes postokted ea~liel fm
mmimi ing fhe ov Ikp legion it clso requi es fuct
bo mda~y conditions ~e specified m two kyels
Repkcmg the spa~se with the dense Lcpkci m will not
effect the mder of fhe scheme, but it will mcke fhe pie-
dictm md cmIectol steps incompabble, so fuct no hue
stecdy state whele bodh equations a~e simultmeously
satisfied c m be leeched The lemedy is Rhie-Chow
interpohtion whele c taggeled tmcge scheme is
cpproximated Dffmting q mtities et conhol vol me
faces by mbsmipt f fhe pledictm equation is explessed
~s
Uf +6t(KU )f ~t(Vu b = Uf AtGfp (23)
Approximate the convectiv md diffusiv temms
by interpohting f om the nodes (ov rbaned telms):
Uf + AtKU AtVu = u AtGfp (24)
OCR for page 915
Note that the pressure values are located on the
grid nodes in this formula. Another approximate
expression is obtained by interpolating the whole of the
predictor equation:
~ ~ ~ n n
u + AtKu - AtVu = u - AtGp (25)
Taking the difference between these gives the
Rhie-Chow formula for the face-centred velocity:
a; = u + AtGpn - AtG pn (26)
Requiring the (face-centred) divergence of u
to be zero gives the pressure equation:
Dfu = 0~AtLp' = Du +At(DG—L)p (27)
where
p' = p*_ pn (28)
The last term in (27) means that no attempt at
removing divergence because of the difference
between the sparse and dense Laplacian will be made if
a steady state is approached. After the velocity compo-
nents and the pressure have been updated the k and £
equations are solved implicitly like in the predictor
step. u is used as the convective velocity.
k + AtKk - AtVk + At—k = (29)
k + Atom
£ + ~tK£ - (30)
~tV£ ~ + AtC —En + ~
82kn
C C k
£n_At 8~ ~ P
In the inner layer the explicit (12) is used for £.
After solving (29) and (30) the eddy viscosity is cal-
culated according to (10) and (13).
On each face of a component grid that coincides
with the boundary of the fluid domain one of the four
boundary conditions given in Table 2 are applied. A
boundary face is identified by the coordinate direction
not tangential to the face, denoted by B. and the direc-
A
tion of the normal B. + 1 if it points into the grid and -1
otherwise. The index B is excluded from the summa-
tion convention. For the continuous problem explicit
boundary conditions are only given for the velocity.
The pressure boundary conditions are derived by
applying the continuity equation on the boundary,
except for the outflow where the pressure is constant.
When applying the continuity equation on the bound-
ary it must be remembered that a staggered storage
scheme is mimicked. The control volume adjacent to
the boundary is half of an interior volume, see the fig-
ure below,
as
The divergence in a boundary point is expressed
Di(U)i = ( 1—~ )2EBB/46i(JVi) + (31)
B(EBB/2 _ l)(Jv )B
The boundary expression for the face-centred
Laplacian is derived like before by applying the diver-
gence operator, now (3 1 ), on the normal flow compo-
nents of the pressure gradient. On all the boundary
types where the Laplacian will be applied, the normal
flow component of u is constant so that the velocity
correction is zero. By this the normal component of the
gradient of the pressure correction on these boundaries
is zero and can be excluded from the expression for the
Laplacian.
Application of the slip and outflow conditions
for the velocities are deferred until the velocity correc-
tion. For the predictor step the Dirichlet boundary con-
dition is used instead.
OCR for page 916
Tetle 2: Bo mdary conditions
Bo mdary Predictor eq Pressure eq
Noslip
u = u
I how Atop' =
Slip
Outdow u = Etu p = 0
Hffe no is the bo mdary surface normal mit
vector The Ne maim bo mdary condition on slip
bo mdaries for the predictor md corrector steps really
only applies to the components tmgentiel to the sur-
face, the nommel component is ffO mpiffmenting f is
directly would couple the component equations, so
instead the normal component of velocity is removed
in c separate step
VALIDATION
During She spring of 1999 the CHAPMAN code
was validated th ough c large m mber of calculations
within the EU-project CAL PSO Mien et cl 1998)
The computed cases included the following:
· Flat pate
· Ellipsoid
· HSVA tardier
· Ry ko Maru t laker
· Modern container ship, two vari mts
· Modern ferry wish flat stern, two vari mts
· Modern ferry wish turmel stern, two varimts
For the plate, the ellipsoid md the Ry ko Maru t laker
the calculations w re carried out bodh et model And full
scale Rey olds m mbffr
The best experimental date available are f cm
the old tmkers, so Other detailed comparison with
computed remits ht. been made for These two cases
Some of These comparisons willbe presentedhere The
reason for incorporating both t miners is feet the HSVA
hull is outstmdmg when it comes to bo mdary layer
md wake measurements et model scale, while the
Ry ko Maru date include remits from th ee R y olds
mnnberi, corresponding to model ma full scale, es
well es m mtemmediete scale More scarce experimen-
tel date have beer amiable for the modem ships, but
some comparison between calculations md meesure-
ments has been possible in all cases U fortunately, the
modern ships are co fidentiel but permission ht. been
g mted to show c few examples f om She container
ship The geomet y of the m odern Kore m hulls used in
the Gothenburg 2000 CFD workshop was not amiable
by the time these celcoktions w re carried out
. ,'.','.','..~.','.','.'W'.'':i U
Figure I Overlapping g id ado md She aft he f of the
HSVA t laker Top: overview, bottom: close-up near
She stern
OCR for page 917
HSVA Tanker
The HSVA tucker is probably the most widely
used test c6 se for ship hydkodynamics CF it we s one
of the cases m 6116 ee workshops on ship viscous
flows held in the eighties 6md the nineties Lesson
1981, La sson et 61 1991, Kodam6 et 61 1994) Refer-
ence is ma de to the workshops for She hull geometry
Only She aft h d of She hull was computed 6md
the g id layout is shown in Figme I Three body fitted
g ids are used to represent the hull 6md its immediate
neighborhood These g ids have 30 points m She direc-
tion out from the hull 6md they are tretched m 6 is
direction using 6 hyperbolic tangent function The dis-
t6 e f om the closest point to She smf6 e is 6pproxi-
mately 6xlO 6 L, which corre ponds to ye 6 o md I
The 6mtomatica By refmed be ckg o md g id is C6 tesiam
with 5 components at 4 levels of g id der sit in the
Figme there is 6 153 6 block surro mdmg the stun where
the der sit has been increased only in the two t ms-
/~W
WE ~—/
:
Figure 2 Compar ison betw en re mlts from the two
blend ratios Top is be ., bottom is .wnkes m the pro-
peller plane
verse di ections Cakchhons w re carried out both
with 6md Echo d this block The results below w re
opts ined with the block included 6md She tots I m mber
of g id points was 289 703 Note that this is for only
the aft he ff of She hull (one side)
Bo md6 y conditions for She cnlcchtions 6 e
given m Table 2 The q mtity up we s obtained f om 6
flat Pete bo md6 y layer solution for the forebodyback
to the i flow station midship This velocity profile
(with am mdisturbed velocity outside She bo md6 y
layer) was 61sc used to mmnhze She solution m the
compnhtionnl domain The same technique was used
for 611 hulls in She validation studies, 616hough the
intention is to Incorporate She new solver in the SH P-
FLOW zoned system La sson et 61 (1990)), Hereby
incre6 sing She 6 curacy of She i flow conditions
The computation was r m with 6 hybrid first
order upwind second order central discretization of
the convective tffmr For She turbulence equations the
blend was 50/50, while for the moment m equations
two blends were tested 20/80 6md 5/95, where She first
m mber corresponds to the first order percentage in
total 180 iterations were made, 90 with 6 time step of
O 1 6md 90 with 6 tep of O 01
In Figme 2 comp6 isons are made between
results from the two blend ratios it is en that She dff-
ferff es 6 e ve y smell, bodh m the pressure distribu-
tion 6md in the isowakes at the propeller plume it is
ml kely that 6 full second order discretization would
OCR for page 918
change the results noticeably.
Comparisons with measurements are shown in
Figure 3 to Figure 5. The pressure distribution along
the waterline and the keel is presented in Figure 4, and
it is seen that the correspondence with the data is quite
good. In this figure and the following x is a dimension-
less coordinate along the hull, with the origin at the FP
and a value of 1.0 at the AP. Limiting streamlines are
shown in Figure 3. Note that the thick lines correspond
to the block boundaries and that the streamlines pass
the boundaries without any distortion. The topology of
the calculated lines is the same as in the oil flow picture
from the measurements, but the regions of upward flow
from the bilge and downward flow from the stern seem
too large. Most likely, this is an effect of the too simple
turbulence model.
.
_ 1 1 1 1
:ss:' ". , l .
0.2~:
0.1
Q
C' O
-0.1
I''---------....,,,, ~
_ \
- i}
Cp-waterline O/
Cp-centreline O /
O Cp-waterline (exp)
~ Cp-centreline (exp) (I/
0!
0/
. .
....
-it
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
x
Figure 4. Pressure distribution along the waterline
and keel
Figure 3. Limiting streamlines. Top: calculations, bottom oil How picture
OCR for page 919
Isowakes are shown at station x = 0.976 (the
propeller plane) in Figure 5. The outermost contours
are relatively well predicted, but the innermost ones are
too smooth, as can be expected with the k-£ turbulence
model.
)). 09_
:~
11
10
9
6
6
5
4
2
1
u1
Otis
0~6
0 1
Figure 5. Isowakes at x = 0.976 (propeller plane).
Top: calculations, bottom: measurements
The cross-flow in the propeller plane is shown
in Figure 6. Thanks to the Cartesian coordinate system
used in both experiments and calculations it is fairly
easy to compare the vector plots. There is, however, a
difference in the reference length for the vectors. In the
experimental data there are two clearly distinguishable
vortices on top of each other and these vortices are seen
also in the computed results, although they are not as
clearly separated.
~~ I_
I t t t t t t t \ \\\\\\\\\\\\
I t t t t t t t \ \ \ \ \ \\\\\\\\
I t t t t t t t \ \ \ \ \ \ \ \ \ \ \ \ \
I I t I t t t t t \ \ \ \ \ \ \ \ \ \ ~ \
'to
/
/`
. ~ ~
'a `:
`.'. ~
~—_ IT,
A; ..
1 ~
',.
.,
,. .
.,
.,
Figure 6. Cross-flow at x = 0.976 (propeller plane).
Top: calculations, bottom: measurements
Ryuko Maru
The Ryuko Maru is one of the test cases recom-
mended by the 22nd ITTC Resistance Committee,
1999. Measured data have been reported by Nami-
matsu and Muraoka (1974) at three different Reynolds
numbers, corresponding to model scale, full scale and
one intermediate scale. Velocity contours are given at
all three scales at a station ~ m (full scale) in front of
the propeller. The reader is referred to the references
OCR for page 920
for a body plan.
~ / /
Figure 7. Overlapping grid around the aft half of the
Ryuko Maru tanker. Top: overview, middle: volume
grids at the stern, bottom: surface grids at the stern
The hull surface is covered with three overlap-
ping surface grids. The foremost grid covers most of
the aft hull and is generated by cutting the surface with
planes of constant x. The grid on the overhang is gener-
ated from constant z cuts. Finally the trailing edge and
lower tip of the skeg is covered with a grid that is gen-
erated with a hyperbolic method, starting at the trailing
edge and marching forward, while following the bot-
tom line.
The surface covered by the foremost grid cuts
the boundary of the computational region at fairly large
angles, so a boundary-fitted volume grid with faces
coinciding with the physical boundaries except at the
rear edge is suitable.
The overhang is more difficult since the inter-
section angle with the symmetry (y=0) plane varies
from large at the water plane, to small down on the
skeg. A grid of the same type as the first is nevertheless
generated, with faces fitted to the symmetry and water
planes. This makes the cells highly skewed in the
region where the overhang blends into the skeg, and
here the tangential resolution is increased by refining
the volume grid twice.
The surface grid at the skeg trailing edge has
two neighbouring sides in the same plane, so if any or
both of the volume grid sides growing out from these
lines were made to follow the symmetry plane, the vol-
ume grid would be singular. Instead the surface grid is
extended out onto the symmetry plane, and a volume
grid is generated from this extended surface. Since it is
only possible to assign a single boundary condition to
each grid face, the volume is split after generation in
three parts, one fitted to the original patch on the hull
and two fitted to the symmetry plane. The two latter
grids form an L around the skeg tip.
All boundary fitted grids are generated with a
hyperbolic method. The grids are the same for all Rey-
nolds numbers, except for the distance to the first grid
point and the number of grid points out from the sur-
face. The number of points in this direction was chosen
so that the grids became approximately equally thick.
| Number of grid points | 47 | 82 | 82 |
ds 1 1.6 10-6 1 5 10-9 1 5 10-9 1
The computational region is then filled with a
coarse Cartesian background grid which is automati-
cally and iteratively refined until a valid overlapping
grid is obtained.
To obtain converged results at the highest Rey-
nolds number a pure first order discretization was used.
This was not necessary at the lower Reynolds numbers,
but for consistency the same discretization was used
throughout. In order to investigate the effects of the
low order a calculation was carried out with a 50/50
blend at the model scale. A comparison with the pure
first order results is made in Figure 8. The differences
are visible, but very small. It was therefore conjectured
that the grid was fine enough to get reliable results also
with the first order method.
OCR for page 921
Figure 8. Comparison between first order and hybrid
first and second order (50/50) results. Pressure dis-
tribution and streamlines on the afterbody
Axial velocity contours at the measurement sta-
tion for the three Reynolds numbers are shown in Fig-
ure 9. Unfortunately the measured region is quite
small, especially at full scale. The thinning of the
boundary layer with increasing Reynolds number is
however apparent, and it is rather well predicted. Note
that the computations were carried out without the pro-
peller in operation during the measurements. This
effect should be rather small, since the measurement
station was about a propeller diameter in front of the
propeller, but it cannot be completely ignored.
The limiting streamlines at the three scales are
shown in Figure 10. As can be expected, the separated
region near the stern is gradually removed when the
Reynolds number is increased. At full scale the separa-
tion seems to have disappeared completely. As the sep-
aration is removed and the boundary layer becomes
thinner the pressure at the stern increases. This is
clearly seen in the figures to the left and means that the
viscous resistance is reduced.
Figure 9. Axial velocity contours at the measure-
ment station 8m (full scale) in front of the propel-
ler. Top: Rn = 7.4x106, middle: Rn = 6.6x107,
bottom: Rn = 2.4xlO9. Port side: calculations, star-
board side: measurements
OCR for page 922
rO.299735
0.270204
0.240674
0.211 1 43
0.181613
0.1 52082
0.1 22552
0.093021 1
0.0634906
0.0339601
0.00442955
-0.0251 01
-0.054631 5
-0.0841 62
n 1 1 In
P
0.299735
0.270204
0.240674
0.211 1 43
0.181613
0.1 52082
0.1 22552
0.0930211
0.0634906
0.0339601
0.00442955
-0.0251 01
-0.054631 5
-0.0841 62
Figure 10. Limiting streamlines and isobars. Top: Rn = 7.4x106, middle: Rn = 6.6x107, bottom: Rn = 2.4x109.
Modern container ship
This hull was one of the test cases in the Euro-
pean project CALYPSO (Tuxen et al 1998~. For confi
dentiality reasons the body plan cannot be shown, but
some results from the stern flow predictions may still
be of interest. Figure 11 shows a close-up of the aft-
most part of the hull with the stern bulb. It is seen that
the surface grids cover the surface very well. One
important detail is the grid on the tip of the bulb. In the
original CAD surface the hole for the propeller shaft
was open, so the surface had to be closed by a cap. This
was generated by a part of an ellipsoid, which was fit-
ted to the edges of the hole. However, since the grid on
the ellipsoid had a singularity at the tip this part was cut
and replaced by a rectangular grid patch.
Figure 11. Surface grids at the stern of the container
ship
The predicted pressure distribution and limiting
OCR for page 923
streamlines are shown in Figure 12. There is an inter-
esting similarity in the topology of the streamlines with
the corresponding plot for the HSVA tanker, i.e. a
branching of the lines near the waterline with a down-
flow near the trailing edge and a separation line where
the flow from the bottom meets the downflow from the
side. This separation line is however much shorter and
has moved to the aft end of the bulb. Much weaker
bilge vortices may be expected from this stern.
.~
~,, , ,,,, ,.: ,
.... .< ,
.. I> ,$ A- ~ ~ ,,.
.. ..... ,.. ,,.
. ~ ,. I. ... ,..
.. ,~.~.~ ,,.
.,. ~,.~,.~,,~f~ ,
. .~ ~.;¢ ~~ ~.~ ~.?, ~ ,
. ..,.,....
..,~> ,,.
~ ,.. I. ~~ ~ ~ .. ,..
Figure 12. Pressure distribution and limiting stream-
lines
Iso-wakes are presented in Figure 13. As
expected both the measured and calculated contours
are rather smooth, indicating that the bilge vortex is
weak. A hook is noted for the innermost measured con-
tour, but this may well be an effect of the shaft that was
present in the measurements, but not in the calcula-
tions. In general, there is a rather good correspondence
between the predictions and the data. For non-vortical
flows this is possible even with the simple k-£ turbu-
lence model.
FUTURE WORK
While the present work has demonstrated the
viability of the overlapping grid method for ship hydro-
dynamic calculations at both model and full scale Rey-
nolds numbers, the full potential of it has not been
used. The overlapping grids used here are best suited to
fully explicit methods, where the updating of the solu-
tion is done on the structured grids completely sepa-
rated from the unstructured updating of the
interpolation points. In contrast to this the requirement
to have an implicit solver that stems both from the
incompressible flow and the thin boundary layers
encountered, forces us to somehow solve the fluid flow
and interpolation equations simultaneously. This is
presently done by assembling sparse coefficient matri-
>.~ ~0 At, ~ - .~ Am, 5, A,
>by. ~ ..,..,. ,..
Figure 13. Calculated and measured iso-wakes in the
propeller disk. Solid line: calculations, dashed line:
measurements
ces from the structured flow equations and the interpo-
lation equations and feeding those to an iterative
solver.
What appears to be the most promising way out
is the Non-Aligned Multigrid method, Johnson (1992~.
Here the smoothing takes place on the component
grids, completely separated from the possibly unstruc-
tured restriction and prolongation operations that han-
dle the inter-grid communication.
The cost for this is that whole grids instead of
only the boundaries of the overlapping regions have to
be interpolated. Experience from the present work has
however showed that the number of interpolation
points is so large that the simpler data structures, arrays
vs. linked lists, possible when interpolating the whole
grids gives approximately the same overhead for the
two approaches.
ACKNOWLEDGEMENTS
A major part of this work was sponsored by the
European Commission under the contract BE95-1721
(the CALYPSO project). The authors are indebted to
Mr. Niklas Wikstrom, who carried out the calculations
for the Ryuko Maru tanker.
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OCR for page 926
M visomwSm
EcoleC nhEled Nmtes,Frdm e
he dmthms present m interesting method for
computing the flow d ound ship sterns bd se d on d
wt of o wrkppmg grids on md d ound fhe hull
F rfd e his work lllushdtes wry cied IY ffhht fhe
flexib le C himf d lik medho do logy which is of f
fmployed to sol w complex flows dhound bodies
wifh dppendEges in motion, mBy also be used with
gf dt profit to reduce fhe discf tisEtion errors by
improVmg the mesh quality m tf ms of
o thogolhllity md refmement in the regions
presenting physical chBllenges if ldenthily, this
workbrmgs d fmal lllustrdtion of th prominent role
pldyed by the turbuief e modellmg for the
prediction of ship stem flows (se fg 2)
When the co figuf d tion which is studied does not
consist of mobile dppendEges, it is dlso possible to
hd w recourse to unstructured grids to impro w the
g id qudiity dS wwll As long dd no f dliY efficie t
un tructuf d grid gef wrdtion tools dh d~Vaildble for
high-Re turbulent flows, fhe use of d set of
o wfldpping structured grids mBybe comidef d dS
mm con~wnient, even if the modern fmite-volume
medhods mBy hedt control volumes of d bitid Y
bSpe Howw wr, the mEm difhculty with fhe
Chimerd -like methods mBy be fhe loss of physicd I
properties such dS mEw md momentum
comf vation, trmsportivity dt the interpoktion
points
Cm you comment on fhe reidti w merits of these
two dpprod hes md indicate if specidl dd-hoc
procedures hd w twen implemented in CH FMAN
to e fof e some of these physicd I properties ?
AUTHOR'S Rb~PLY
Ovuall we beheve fh q all viable m hods w 11 give ~ milv resclU
he mm differmce bflweec fhe methods h fhal pqIal
dhmele qbC h emier oc a smooth qmclmed gnd especiallf if a
high order of a mra f h deshed, hLe comerv hOS h easier lo
mfome os ag idwithcos overlappkg celh
hhUI wvdspefechg wul ppkggrlds
Mosllf mooth bosad f, so fhal f w c mpocecl grids aw
wqched ald fhe rel qlve eff mpecl oc h I mol qloms Iow
High Re reqchhg high ~pe t~aho celh Se h g ids aw
emler lo gecuale m ~ pvale mclmed c mposecl g ids
No shochs, so the whhoc cm be rewlved ald the h a mrav
matlf cm probablf)teceglecled
For fhe case of rlgld bodies mwhg relqlve lo ose mother, d
ppears fhat d ts more ehiclent to ~se a fh ed R id ~odad ea h bodv
fhh case moq of fhe ma hh f reqched h alreadf pesml m al
werl pph g grid m hod, while addh g fhh capabllllf lo m
m~bmclmed grid method I hes more woh
Representative terms from entire chapter:
pressure distribution
