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OCR for page 762
The Numerical Simulation of Ship Waves
Using Cartesian Grid Methods
M. Sussman (Florida State University, USA)
D. Dommermuth (Science Applications International Corporation, USA)
Abstract
Two different cartesim-g id methods are used to simu-
late She flow aro md the DDG 5415 The -i st tech iq~x
uses c "coupled level-set md vol me-of-fluid" (CLS)
techmique to model the flee-surface i te face The no-
flux bo mdary c Edition on the hull is imposed usi g c
flmite-vol me tech iq~x Th second techmique uses c
level-set techmique LS) to mod I he fiee-surface inter-
face A body-fmce tech iq~x is used to impose She hull
bo mdary condition The predictions of bodh m meri-
ccl tech iq~xs are compared to whisk r-probe mecsure-
ments of She DDG 5415 The level-set tech iq~x is also
used to investigate the breakup of c two-dimffmsiorul
spmy sheet
1 Introtlttcbott
At moderate to high sped, She turbulent flow along
the hull of c ship md behind the stem is characterized
by complex physical processes which involve break-
ing waves, air entrainment, fiee-surface turbulence,
md She formation of prey Irsdmonal m meri al m-
proahes to these problems, which use bo mdary-fltted
g ids, me di flcult md time-consummg to impleme t
Also, es wakes steepen, bo mdary-fltted g ids will Ixek
d wn Bless cd hoc h ectments me implemented to pre-
ve t fhe wa~s from g tting too steep At fhe wxy
lest, c bridge is req ired betw en potenticl-fl w medh-
ods, which mod I limited physics, md more c mplex
bo mdary-fltted g id methods, which mcorpomte mme
physics, clbeit wifh g ect effo t md with limitations on
the wave steepness Cartesi m-g id methods me c m~t-
ural choice be mse fhey all w mme compl:x physics
thm potential-fl w medhods, md, unlik bo mdary-
fltted medhods, cartesi m-g id methods ~equi~e minimal
effo t wifh no limitation on the wave teepness Al-
though cartesim-g id methods (CGM are p~esently m-
ccpable of resolving fhe hull bo mdary-kyer, CGM c m
model wa~ b~eckmg, fre-smface tmbulence, cir ff~-
t~aimment, spmy-sheet fommation, md complex interac-
tions betwen the ship hull md the free surface, such
es t msom-stern fl ws md t mblehome bows The
cartesi m-g id methods fnat are descobed m fhis pcper
use the pmelized ge meby fnat is used by potenticl-
flow medhods to cutomatically con truct c represent~
tion of the hull The hulMepresentation is fhen im-
mersed inside c cartesim g id that used to tmck fhe
interface No cdditiorul g iddi g beyond what is cl-
redy used by potenticl-fl w medhods is ~equi cd We
note that mother variction of this cpprocch is to use
cartesim-g id medhods to hack the fiee-surface inter-
face md body-fltted g ids to m odel the ship hull
For the cclcoktion of ship waves, VOF md level-
set methods have cc tam cdvmtages md disadvmtages
VOF uses the vol me fraction (F) to h cck fhe mterface
F = 0 corre ponds to gas md F = I corre ponds to
liquid For intermedicte values, betw en ero md one,
there :xists m interface betw en the gas md the liquid
The interface betw en the gas md the liquid is sharp
for c pure VOF medhod L vel-set methods use c level-
set f mction (f) to model fhe gas-liquid interface By
deflmition, f 0 dffmtes liquid,
md f = 0 is fhe mte face For conventiom~l level-set
sch mes, the mte face betw en fhe gas md th liq id
is given c flmite thickmess[17], which is ml kc conven-
tiom~l VOF schemes[3]
In the case of fre-smface fl ws, where fhe dff~-
sity rctio betw en cir md water is ckmost f ee orders
OCR for page 763
of mug itude, She Smite thick ess of She Ate fan that
charateries level-set methods hr. two cdv Stages own
VOF First, the Smite Hick ens tends to smooth j mps
in the t mgenticl component of the velocity on She m-
terfae Second, the Smite Hick ens tends to facilitate
using multig id methods to solve various types of ellip-
tic equations that involve the dew it
The ad ection algorithm that is used for VOF con-
serves mass if She flow field is solenoidal The level-
set advection equation tends to sac mohte m merical
enors For the level-set method, She level-set func-
tion must be periodically reinitialized to mcintam c
proper Hick ens for She interfae, of herwise the inter-
fae would become either too thick or too thin The
reinitialization process is c sig ifl It source of errors
in She level-set medhod Based on acuray consider-
ctions, She calcoktion of g ~~ -d i en -I ws tends to
favor VOF over level-set methods
The interfae is neconshucted fiom the vol me fra-
tions m VOF Dm ing the recon me non process, She m-
terfae nommcls Ed curvature are calculated Typically,
the calcoktion of the i te face nommcl ad curvature me
less accurate for VOF f m for level-set medhods The
interfae nommcls Ed curvature are c tlcrd.tted dinectly
in level-set methods m temms of g cdients of She level-
set function As c result, She cclcuhtion of the nommcls
Ed curvature me less costly for level-set methods rela-
tive to VOF The cclcohtion of smfa tension effects,
which are c function of the curvature of the mterfae,
tends to favor level-set methods own VOF due to con-
siderctions of acuray Ed efficiency
O ha retched multidimensional g ids, VOF
methods are less prone to clicsmg errors f m level-set
methods Level-set methods incur errors as the interfae
rotates th o gh highly resolved regions into regions that
me not reach ed w 11 This t pe of clicsmg error occurs
in cartesia-g id methods when She mesh along one co-
ordincte axis is m me finely resolved th m clo g mother
coordinate axis
By deflmition, the level-set Ed vol me-of-fluid func-
tionbothall wmixmgofgas mdliquid his legate of
level-set Ed vol me-of-fluid methods may be desi Cole
for modeling gas emrai merit mch as She air Nat is ff~-
t~ained by c t rat mg wave During She ~einiticlization
process, level-set methods Ed "coupled level set Ed
vol me-of-fluid methods" (CLS) use c sig ed dist Ice
f motion to update the level-set f motion Ed the fhick-
ness of the ate 19c Nctmally, She di tance f motion
could be used to model She intensity of turbulence Ed
limo It of gas enbai merit es c function of the did Ice
to the ate 19c
Dommermuth, et cl, (1998) used c trctifled -I w for-
muhtion to simulate t ret mg bow waves on She DDG
5415 et c Froude n mber Fr=0 41 Thei m merical
results compared well to whisk r-probe mouser merits
in the b w region [8] H wever, Dommermuth, et al,
(1998) identified two issues that requited further st dy
First, thei tmtifled -I w formulation allow d the free-
smfa interfae to become too diffuse Second, She
c outact -l me h ectm ent di d ot cl low She tree surface t o
rise Id fall cle mly along She side of She hull The two
new m mericcl tppro tohes that are discussed m d is pa.
per are attempts to h m edv these problem s
Both m meri 91 cpproah s use c sig ed distmce
f motion to represent She hull The dist m. e of c pomt
to She hull is negative Aside She hull Id positive out-
side She hull The flmite-vol me tpprotch uses She
sig ed dista e to calculate She area md vol me fra-
tions for computational cells cut by the hull, whereas
the body-force technique uses th sigmed distmce to
prescribe a sm ooth facing term The coupled Ate fae-
trsckmg algorithm (CLS) uses level-set to oslcokte She
normals land curvature if needed) to She fi ee-surfae m-
terfae that are used m VOF The sdvection po tion of
the slgorid m is pe fommed by VOF [16] The level-
set interfae-hscking slgorid m uses a new isoturfae
sch me to calculate She zm3 levels t Then th mini-
mal distance betw en She cartesim pomts ad the zero
levels t is calculated in a narr w bad The minimal
di tance is made positive m the water ad negative in
the sit This sigmed dista e to the free surface is used
to reimtiali:D: She thickmess of She interfae
The two m merical reproaches ah used to simulate
the flow pro Ed the DDG 5415 The CLS technique is
still mder development, so only preliminary results ah
presented The level-set technique inchdes upg odes to
the m meri 91 technique that is det Ned m [8] Those
upg Ides include a new body-fcrce formulation dint is
mollified, a new h initialization procedure, md a new
flmite-vol me hestment of the convective temms The
original m merical procedure is not mollified md does
not use remitislizati m in addition, She original cenbal-
difference formulation of She convective terms is not as
robust as the new to stment using a flux integ al for-
mulstion We -i st revi w the governing equation md
then we die uss She m meri 91 spproahes Finally, w
present some pa lim inary m merical results which illus-
trste various festmes of the m merical slgorid ms The
application of level-set methods to She be tkup of spray
sheets is also illu orated
2 Field Equations
As in Dommermuth et al, (1998), consider turbulent
flow at She interfae betw en sir md water [8] L t u,
denote the d ee-dimensionsl velocity field as a fi motion
of space (a\) md time (t) For a incomph sable flow,
OCR for page 764
the conservation of mass gives
~ off) = i+(l i)H(f)
5~ =0 (I) Off) = :+(l :)H(f), (6)
u, Ed a\ me normalized by TO Ed Lo, which are She
characteri tic velocity Ed length scales of the body, re-
spectively On She smface of the m ovmg body (Sb), She
fluid particles mm with thebody
Ul = A\,
(2)
where A\ is the velocity of She body
Let V: Ed Vg respectively denote She liquid (water)
Ed gas (pi ) volumes Foll wing z procedme that is
similar to [13, 15], w let f denote z 1evel-set f motion
By deflmiti m, f = I for x ~ Vg Ed f = I for x ~ V:
he fluid interface cmTe pond to f = 0
he cow tion of f is expressed as follows
Of 5Q
at OR '
(3)
where d/dt = 5/5t + u,S/Sr, is z s bet mtial derive
me ;? is z mb-g id-scale -I x which c m model She
enhai merit of gas into She liquid D tails me provided
in [8]
Let Pt Ed pi, respectively denote She dew it Ed dy-
rumic viscosity of water Similarly, pg Ed Us are She
conespondmg prope ties of pi he -I w in the water
Ed pi is governed by the Na vier-Stokes equations
= F\ ~ + p ~ (2kS\~)
F ;'; + W T\ + ~ V, (4)
where to = P:UoLo/p: is She hey olds number, F9 =
U9/(gLO) is6heFroudemmber, mdW~ = prU2Lo/o
is the Weber number 9 is the acceleration of g avity,
Ed o is She smface tension F\ is z body force At is
used to impose boundary conditions on th sm face of
the body P is She pressm e T\ accounts for smface
tension effects i\> is She Kronecker delta symbol As
described in [8], TV is the mbg id-scale shess tenser
S\> is She deformation tensor
I 49U\ + S~N (5)
2 :~> Aid
md ~ oecti ok th dimnn~inulem~nri~hlnden-
where ~ = Pg/Pt Ed ~ = kg/~t me She density Ed
viscosity ratios betw en air Ed water For z sharp m-
terface, wish no mixing of air Ed water, 11 is z step
function in practice, z mollified step function is used
to provide z smooth h man on h ens een air Ed water
Based on [3, 4], the effects of sm face tension are x-
pressed 15 z smg lar somce term m the Navier-Stok s
equations
(f)35 (f)
C)
pressed m temms of the level-set function
Off) = V ( Vf )
he pi e s sm is reformulate d to zip sorb She hyd o -
static term
(8)
(9)
where Pd is the dynamic pressme Ed Pa is z hyd o-
static pressm e temm
= J dz PA )F (10)
As discussed m [8], the divergence of the momentum
equations (4) in combination with She conservation of
mass (1) provides z Poisson equation for the dynamic
pressm e
~ I SPa
go\ P be\
the velocity onto z solenoidal field
(11)
whereZ is z somceterm Equation II isusedto moiect
3 Enforcement of Body Bounditry
Conditions
Two different cartesi m-g id methods are used to simu-
late She flow arommd the DDG 5415 The -i st tech iqm~
imposes She no-fl x boundary condition on She body us-
ing z flmitewolmme tech iqm~ The second tech iqm~
OCR for page 765
imposes the no-fl x bo mdary conditi m viz m exterrul
for e field Bodh techmiques use z sig cd dist mce func-
tion <6 to r present the body <6 is positive outside 6he
body md n gative mside the body he mag it de of <6
is 6he mmimal dist mce betwe n 6he position of <6 md
the surfae of the body
With r spect to the vol me of fl id 6~t is en losed
by 6he body (Vb), w deflme z f mction i:
I for x ~ Vb
i(x) = ~ for x ~ Sb
O for x ~ Vb
(12)
he function ~ mbe expr ssed in terms of z surfae
di trib xion of normal dipoles [ 10]
i(x) = 4 J dz'5, t, (13)
where n is 6he outward-pomtmg mit normal to 6he
body, md t is z F~mkin source, t = x x <6
is expr ssed m terms of ~ zs follows,
¢(x) = i(x) x x m~,
where x x m\~ is 6he mmimal dist mce between 6he
field pomt x md 6he points on the body x in pra-
tice, the body is discr tized using h i mg lar panels As
z result, the calcoktion of the minimal di tance sweps
over zll the h i mgles comprising 6he body md must a-
co mt for th possibility 6~t the minimal dist mce may
occur eith r zt the cor rs of h i mgle, zlong 6he edges
of h i mgle, or im ide 6he h i mgle
3.1 Free-slip conditions
(1 4)
In the flmite vol me zpproah, 6he i reg lar bo mdary
(i e ship hull) is represented m temms of <6 zlong with
thecornspondmgarefmctionsA mdvol mefrations
V V = I for computational elements f ily outside 6he
body md V = 0 for c mputational cl ments f 11y m-
side the body he repr sentation of ineg lar b md-
aries viz arez frations md vol me fiations has been
used pr viously in 6he foil wmg work for mcompress-
ible fl ws [I, 19, 5]
Recall 6he pressure equation,
V Vp = V W
wi6h the following no-flow bo mdary condition:
nW~z = W nw~n
(1 6)
where nw~z is 6he outward normal d zwn from 6he a-
tive flow region mto 6he geomet y region
For eah discrete computational element 14~
deflme 6he geomet y vol me fiation V md arez fra-
tion A zs
~\>u Jnw ~
A\+r/25U = r J El(~)d~
\+~/2,~,u r +~
F\+r/2,~,u r presents the lef fae of z comput~
tional element; similar deflmitions zpply to r\ r/2 ~ u
r\,>+~/2,u.
In or der t o discr te Iy e f orce 6he b o mdary c ondit ions
(16) zt 6he geomet y smfae, w use z flmite vol me
zpproah for discretizmg (15)
Given m i reg lar computational element 14>u (see
Figme 1), w have
J v UdV= J u nw~ dA
~w~ b~w~
he divergen e 6heor m motivates 6he followi g second
order mproximation of 6he divergen e V U zt 6he c n-
troid of l:~u
v u~ ~4 J U nw~zdA (17)
~u b~w~
In temms of geomet y vol me frations V~u md mez
fmctions A\+r/2~u, (17)becomes,
V U~ V u~Y~Z[
(A\+~72,~,uAyAz)~\+~/2~,u
(A\ t/2,~ YZ~Z)~\ t/2,~,u +
(A\~+~/2,uA~Az)v\,>+~/2,u
(A\~ t/2~ Z)v\~ t/2,u +
(A\~,u+~/2~y)w\~u+~/2
(A\~,U t/2~y)w\~u ti2
L~U"U~ ]
For z :D:ro fl x bo mdary condition zt 6he wall, the h t
term m (18), LW~tiU~ll nw~z, is ~
7he flmite volume zpproah, when zpplied to 6he di-
(15) vergen operatorm(lS)becomes:
V pVP~ V uA~AYAz[
A\+~/2~uAyAz(p~lp)\+~/2~
A\ t/2,~ YZ~Z P~/P)\ t/2~,u +
(1 8)
OCR for page 766
/
md
A\~+~/2~uA~A3(py/p)\~+~/2~u
A\,> t/2,uZ~ z(Py/P)\,> t/2,u +
A\,~,u+~/2~Ay P2/P)\,~,u+~/2
A\,~,U t/2~AY(P2iP)\~u ti2
L f Ul'(VP/P) f U'' nw~]
V W~ V UA~AYA3[
(A\+~72,~,uAyAz)~\+~/2~,u
(A\ t/2,~,uAYA3)~\ t/2,~,U +
(A\~+~/2,uA~Az)v\,>+~/2,u
(A\~ t/2~u~Az)v\~ t/2,U +
(A\~,u+~/2~y)w\~u+~/2
(A\~,u t/2~y)w\,~,u t/2
L f U66W f u nw~]
D e to 6he no flow condition (16), the terms
L f U'l(Vp/p) f Ull nW~z md L f UllW f U6l nW~II cancel
each othe' The resultmg discretization for p is:
A\+~/2~uAyA3(p~/p)\+~/2~
A\ t/2,~ YZ~z(p~ip)\ t/2,~,u +
A\~+~/2~uA~A3(py/p)\~+~/2~u
A\~ t/2,uZ~ z PY/P)\~ t/2,u +
A\,~,U+~72~Y(P2/P)\~,U+~/2
A\~,U r/2~AY(P2/P)\~,u ri2=
(A\+~/2,~,uAyAz)~\+~/2~
(A\ t/2,~,u~Y~Z)~\ t/2,~,U +
(A\,>+~/2,uA~Az)v\,>+~/2,u
(A\,> t/2~u~z)v\~> t/2,u +
(A\,~,u+~/2~y)w\~u+~/2
(A\,~,u t/2~y)w\,~,u t/2
where, for example, (P~)\+~/2,~,U is dismetized zs
p\+~u p\~u
3.2 No-slip conditions
The bo mdary condition on the body c m zlso be im-
posed using m external force field Based on Dommer-
m xh, et zl, (1998), 6he di tance f mction rep~esentation
of 6he body (~6) is used to consh uct z body force as fol-
I ws:
¢<0 \
A\,>+ry2 ~ 1/~
~'
/~>0 f,j
V\> ~ 7/8 A\+~72~ = 1
A\ t/2~1/2
A\,> r/2 = 1
Figme 1: Diag zm of computational element ( f, j) that
is cut by 6he embedded ge meby
F\(x,t) = cfA(t) (I exp (
46(x)l~) ))~\(x,t)
V <6(x) < 0, (19)
where cf is z fi iction coeffcient ~ is used to mollffy
the body force mch 6~t it is g zdually zpplied across
the surface of the bod Recall that <6(x) < 0 corre-
sponds to points wi6hin th body F\ = 0 outside of 6he
body A(t) is m zdju tment f mction:
A(t) = 10 exp( (t/To))
(20)
To is the zdjustment time The zdju tment f mction
smoothly in eases to mity from its initial value of zero
The effect of 6he zdju tment f mction is descobed in [8]
The zdjustment f mction reduces th gen mtion of non-
physical high-frequen y waves
As constn ted, the velocities of 6he pomts withm 6he
body are forced to vmo For z body 6~t is fixed in z fiee
sheam, 6his corre ponds to imposmg no-slip bo mdary
conditions
4 Interface Tracking
Two methods me presented m our work for comput-
ing ship fl ws Both medhods use z "fiont-captmmg"
typ procedme for representmg 6he flee surface sep~
rating the zi md water The fl st techmique is based
on 6he Coupled vol me-of-fluid md level set medhod
(CLS) md 6he second techmique is based on 6he level
set method LS) zlon
OCR for page 767
4.1 CLS method
In this section, we describe She Ed coupled L vel Set md
Vol me of Fhid (CLS) clgori6 m for replesenting She
flee surface For mme details, e g axisymmetric md
3d impl mentations, see [16] in She CLS algorithm, She
position of She mte fade is updated th ough She level set
equation(levelsetf mctiond notedbyf\~) mdvol me
of fluid equation (vol me flation of liquid withm each
cell is denotedby Fv ),
f2 + v (up Of) = 0
F2 + V (U~ACF) = 0
In order to implement She CLS clgori6 m, w me gn en c
discretely divergence fiee velocity field u Add deflmed
on She cell faces (SAC g id),
a,+/ a\ t/2~ + V,~+1/2 v\,> ./2 0 21)
Given f A, F f md ISAAC, we use c "coupled" sec-
ond order conservative operator split cdvection scheme
in order to fled ff+t md F~+t he Ed operator plit
algorithm for c g nercl scalar ~ foll ws es
i ~ f + ,,~(G\ t/2,> G,+l~2~) 22)
1 ,,'(U,+1/2~ a\ t/2,>)
s~+t = TV + ~ (G\,> ~/2 G~+l/2)+
;v(v\~+~/2 :., t/2), 23)
where G,+l/2d = s\+~/2,~u\+~/2~> d notes the flux
of ~ across the right edge of She (I, j)6h cell md
G \,>+ ~/2 = f \,>+ 1/2V,~+1/2 denotes She flux across She
top edge of the (I, j)6h cell he operations 22) md
(23) represent the case when one has She ;- w ep" fol-
lowed by She "y- weep" Al ter every time step the order
is reversed; -x-- w ep" (done implicitly) follow d by She
"x-swep" (done explicitly)
he scalar flux s\+~/2,> is computed differently de
pending on whether ~ represents She level set function f
or She vol me fraction F
For She case when ~ represents the level set func-
tion f w have the foll wing representation for s\+~/2,>
(U\+~/2,) > 0),
s\+~/2,> = ~ f + 2 (Dim f +
where
he clove discretization is m otivated by the second or-
der predictor conector method described m [2] md She
references 6herem
For the case when ~ represents the vol me flation
F w have She foll wing representation for 9,+1/2d
(U\+l72,) > 0),
where
(I' (~,y))d~
6\+~/2,> u\+1/2~AtAY
24)
= (MY) ~\+~/2 u\+~/2~t < ~ < ~\+~/2
ad y: r/2 < Y < Ys+l/2}
he mteg cl in (24) is evaluated by flmding the vol me
cut out of th region of integ ction by the Ime repre-
sented by She :D:ro level set of f FIR
he term f ~ (A y) fo md m (24) represents -. Im
ear rec instruction of the mte fade m cell (I, j) ~ other
words, f AIR (v y) has She fomm
f,f (a, Y) = a\> (A A\) + by (Y Y>) + C\> 25)
A simple choice for the coefficients c\> md by is es
foll _ >,
to= 2~(f\+t,> f\ i,>)
a\> = 2 y(f\~+t f\> I)
26)
27)
The intercept CV is determined so Hut She Ime repre
seated by the zero level set of (25) c Is out She same
vol me m cell (l, j) es specified by F\> in other words,
the foll wing equation is solved for cv,
JO ET(~V(~ a\) + tV(y y>) + c\~)d~ Din
where
( U\+l72,>(D_
(Dim f = \+t,> \ i,>
~9) >)
Arty
= (MY) Hi t/2 < ~ < ~\+~/2
md y: ./2 < Y < ys+r/2}
After f ~+t ad F~+l have been updated tccordmg
to (22) md (23) w "co pie" She level set f motion to
the vol me Irtcnons as a part of She let I set remitisl-
ization step The let I set remiritliztrion Rep reply s
the current value of f~+t with She exact dista e to
the VOF recon erected interface At She s tme time, She
VOF recon erected mte Lace us s the current value of
f ~+t to determine the slopes of the pie ewit linear re-
conshucted mte la
remarks:
OCR for page 768
o he dista e is only eded in c tube of K cells
wide K = 6/~r + 2, the~efore, we c m use "brute
force" techmiqu s for flmding 6he exat dista e
See [16] for detsils
o Dm mg the ~einitislisation step we h u ate 6he vol-
ume fiations to be O or I if f > ~r A16hough
w hu cte 6he volume flations, w till observe
6~t mass is conserved to withm c flation of c per-
cent for ou test probl ms
4.1.1 CLS Contact angle boundary eondfltions in
general geometries
he CLS contat mgle boumduy conditions me ff~-
fmced by extending f i to ~egions where V\> < I (i e
initisli mg "ghost" valu s of f m the inactive portion of
the compubtiorud domcm)
he conbct mgle boumdary condition et solid walls
is given by
n nw~ll = cos(~),
28)
where ~ is c user deflmed conbct mgle ad nW~zz is 6he
outward normal d awn from 6he ative fl w region i to
the geometry ~egion
In terms of f (the fiee surfae level set fu tion) md
<6 (the geomeby level set function), (28) becomes
Vf V<6 = cos(~)
In figme 2, we sh w c diag sm of h w the conbct mgle
~ is deflmed in terms of h w 6he fre su fae mtersects
th geometry surfae
Solid
\~ ~ Gas
.~
Liquid
Figme 2: Diagrsm of gas/liquid mterfae meetingat6he
solid Ibe dashed Ime represents the imcgi uy inter-
fae mected th u 6he level-set extff~sion procedme
he "extension" equstion hcs 6he form of m cdvec-
tion equstion:
f T + U Vf = 0 <6 < 0 29)
In ~egions where 0, f is left umch mged
For c 90 deg ee contat mgle (6he defalt for ou
computations), w have
~ ~d V
V:
In other words, i formation propagates nommcl to 6he
geomet y su fae
For contat mgles dfffe~ent fi om 90 deg ees, 6he fol-
I wing procedu e is tsken to flmd ue~ d:
Vf
Vf
V:
nw~ll = Vi,,
n~ = n x nw~
n x nw~ll
nI x nw~n
n2 = nI x nw~n
c = n n2
Remarks:
~ n~,, GOfl1 8)n2 ffc < 0
~ d I n~,, GOt c~n2
., = ~ ~,,+GOI Wr ff c > 0
I nw~+cot ~ s~n2
~ nw~z ffc = 0
o ~ 3d, the contat line (CL) is the 2d cu ve which
~epresents the mtersection of the fre su fae with
6he geomeby su fae (ship hull) he vector n2 is
o thogomd to the contat line (CL) md lies in 6he
tmgent phne of the geomeby su fae
o Smce bodh f md <6 me deflmed withm c narrow
b md of the :osro level set of f, we c m also deflme
~ "d wi6hin c narr w b md of 6he flee su fa
o We use c fl st order upwind procedu e for solvmg
29) he di ection of upwmding is detemmmed
fiomtheextensionvelocityu a ~ d Wesolve 29)
for r = 0 6
o For viscous flows, there is c co flict betw en 6he
no-slip condition ad the idec of c movmg contat
line See [6, 8, 9, 12] md the ~efe~ences 6herein
for c discussion of this issu We have pe formed
m mericcl tudies for axisymmeh ic oil precdmg
m water umder ice [18] wi6h good cg cement with
experiments in the futme, w wish to experiment
wi6h cppropriste slipboumdary conditions near 6he
contsct Ime
4.2 Level-set method
A k y part of level-set methods is remiticlization Wi6h-
out reinitislisation, the thick ess of 6he mte fae be-
tw en the gas md the liquid ca get ei6her too 6hick or
OCR for page 769
too thm Reinitialization is based on the consh uction of
z sig ed distance f mction that rep~esents 6he di tance of
pomts fiom 6he gas-liquid mte fae By defimition, 6he
sig ed dist mce is positive m 6he liquid md n gative in
the gas At the interface, the dist mce f mction is zero
A variety of methods have be n utilized for calcuht-
ingthe sig ed di tance f mction in hdingahyperbolic
equation [17] md di ect methods [16] The hyperbolic
equation medhods tend to be less accumte but mme ef-
ficient 6 m di~ect method Here, w outlin z di ect
method 6~t c m be eff ciently implemented on parallel
computers with second-order acuracy The m merical
sch me c m zlso be gen ralized to higher orde'
First, calcokte the mtersection pomts (xg) where 6he
:D:ro level-set crosses each of the cartesi m ~s At
these mtersection points calcuhte 6he nommal to 6he m-
terface (np) Togedher, xg md np determme local m-
proximations to 6he p mes that pass th ough 6he zero
level-set For pomts 6~t me within z nnrow b md
of these p mes, calcokte th minimal distance to 6he
planes Once th minimal di tance is calcohted, zssig
the sig of th dist mce function based on 6he sig of 6he
level-set f mction
For example, consider z zero crossmg zlong 6he
z xis Locally, n ar the zero crossing, 6he level-set
f mction f is fitted with Lzg mge polynomials
u=K
f(zo) = ~ Lu(zo)fu ~
(30)
where zO is offset where mtemohted level-set func-
tion f = 0 Lu are Lzg mge polynomials md fU ~ e
discrete values of f n ar the :D:ro level-set zlong 6he
z xis K I is the deg ee of the interpoktmg poly-
nomial zO is calcokted di~ectly for I w-order polyno-
mials md iteratively for high-order polynomials Let
x0 = (~0, yO zO), where (~0, yO, zO) is the coordin~te
of 6he :D:ro crossing
The mit normal nO zt the zero crossing is calcohted
in terms of the level-set function:
n0 = V f zt x = xo ,
where 6he g zdient temms me calcohted usmg fimite dif-
feren e formohs of desi ed crde'
The minimal dist mce (z) betw en z pomt (xp) md z
plane lies zlong 6he mit nommal to the p me Denote 6he
position where 6he point intersection occurs zs xs, 6hen
where z is exp~essed m terms of z dot product:
(31)
(32)
Z = (xO Xp) nO
(33)
Note that higher-order conections mvolve curvatme
terms, etc As long zs xO xs < Ag, where Ag is
the g id si:D:, then g is potentially 6he minimal distance
to 6he :D:ro level-set 06her c mdidates in hde planes
in the n ighborhood of xp on z shuctmed g id, shffts
zlong the cartesi m ~s m be p rfommed to consider
oth r c mdidates only xp n ar the :D:ro level-set me
required m 6he ~einitialization procedme A simple pro-
cedme for fimdmg points n ar 6he vmo level-set in lves
w ighted a~rages Fir t conshuct z stai -case zpproxi-
mation (~) to the zero level-set:
t\,~,u = I v f\,~,u > 0
t\,~,u = 1 v f\~,u < 0 (34)
A weighted average along 6he k-6h indice is
T\,~,u = (~\,~,u+t + T\~,u + T\~,u t)13 (35)
Simibrexp~essionsholdalong6hef th mdj thm-
dices Repeated zpplications of weighted avemges pro-
vide z nnr w b md that en ompasses 6he :D:ro level-set
Thenarrowbmdcorre pondstoth ~egion T\,~,u < I
The sig ed di tance f mction D is expressed in terms of
th level-set f mcti m md the mimmal distance:
D = sign(f)g
Bzsed on [17], ll(f) is ~einitializedasfoll ws:
iT(f) =
ET(f) = sin(
(36)
~ ) if D < Zi
II(f) = I if D < A, (37)
where ~ is 6he desired thickness of the interface
5 Flux Integrt~ Methods
We defin 6he tempmal md patial a~ragmg over z
time step md z cell zs follows:
~ 1 2+~2
f = ~t~ v `.l dt J dv f (38)
where he~e, the tilde md overbar symbols ~espectively
denote temporal md spatial avemging At is 6he time
step, md AV is 6he vol me of th cell
OCR for page 770
As m example, consider the application of She pre-
cedmg operator to th level-set equation (3):
f f + Off 5Q
't :~ go,
(39)
where here superscript n denotes the time level We
focus on attention on She come me term he convec-
tive term accounts for the flu of She level-set f motion
across the faces of the conk ol volume A second-order
approximation for She flu am oss one face of z cell is
providedbel _:
JO JY2 J22
a, yl z.
where F+ is She flu across the positive face along She
x axis he lim its of integ ztion are provided below:
~~ =
2
= 2 7 +At
Ay A~ v
2 ( 2 ) 7 +
= ( 2
fez A~ w
2 ( 2 )7+
A~ w+
+(~ 2 )7+ '
32 =
(41)
where An, Ay, md Az me She lengths of the cell along
the cartesi m 9:~t. 7 + is She normal component of fluid
velocity It the center of positive face along She ~ axis
v+ md v are She normal components of She fluid ve-
locities It She centers of She positive md negative faces
along the y axis Similar deflmitions hold for w+ md
w
In z m mped c cord wee system, She expression for She
flu is
F+ = / / / drdzdt J f(~,y,z), (42)
where J is She Jacobi m, md a, y, md z me f motions of
r,z, mdt:
Y ~
= 2 ( 2 )7+At
zany (1 + r)(l + Z)V
2
(I + v)(1 z)v At
4
t (I + v)(1 + t)w+Z:.t
2 4
(1 + v)(1 t)w Z:.t
For this partic jar zpproximation,6he Jacobi m is
J S~ y Sz
S~ is St
(44)
On my one face the stencil associated wish the Lt-
g mgi m interpolation of f is 3 x 3 x 3 = 27 points,
but for She enti e cell, the stencil is 5 x 5 x 5 = 125
pouts We use z upwindbiased Pencil for the mo-
mentum equations md z symmeh ic stencil for She level-
set f motion he diagonal md cross terms in the mo-
mentum equations are heated the same Generally, w
use eight-pomt Gaussi m quad zture to evaluate the flu
over each face Details of She m merical zlgori6 m me
described in [7] Various t pes of limiters are described
in[11]
6 Preliminarv Results
In section 6 1, w prese t prelimi By computations
of -I w past z DDG 5415 ship ~ section 62, w
present preliminary computations of She breakup Of Z
two-dimensiomtl prayshet
6.1 Ship Wave Results
As z demonstration of She level-set md She coupled
level-set md volume-of-fluid fcrmnlations, we pre-
dict the flee-su face di tu bance near She b w of She
DDG 5415 moving with forward speed The experi-
ments w re performed It She David Taylor Model Basin
DTMB), md me available viz the world wide web at
http ://wwwSO dt ma y m ii/5415/ This is the same flow
that Dommermuth, et al, (1 998) originally investigated
using thei shatifled flow formulation [8] As before,
w only consider She high speed case For this case, z
plumbing breaker fomms near the bow Ai is ennui ed
md splash up occurs where the _zr7 reenters the flee
surface There is flow separation It She stern, md She
t msom is d A large rooster tail forms just behind
the stern
Based on She speed (Uo=6 02Knots) md She lengh
(Lo=5 72m) of She model, She Rey olds md Froude
numbers are to = 1 8 x 107 md Fr2 = 0 41 The
effects of surface tension are not included The density
ratio of air md water is ~ = 0 0012 md She ratio of She
dynamic viscosities is g = 0 018
OCR for page 771
In regard to the m merical parameters for the level-
set fcrmoktion, w use z friction coefficient Of = 500
in She body-force temm (19) The adjustment time is
To = 0 02 For She level-set fommoktion, the length
md width of the comp tatiorurl domain are L = 2 5
md W = 1 50 The h ight of She air above the me m
fiee-surfae is h = 0 15 md She depthbel w th me m
fiee-surfae is d = I O. One g id resolution is used
with 512 x 128 x 129 g idpoi Is Th e deferent levels
of g id shetchi g are used along She y md z as
For She highe t g id resol lion, the smallest g id spa-
i gis26xlO 3zlongthey axis md36xlO 9along
the z axis For She medi m resolution simulation, She
smallest g id sprung is 3 8 x 10 3 zlong She y axis
md 1 8 x 10 3 zlong the z axis For the coarse t g id
simulation, the malle t g id Spacing is 3 8x 10 3 zlong
the y axis md 3 5 x 10 3 zlong the z axis The g id
Spacing (4 9 x 10 3) is con t mt along the ~ axis for
all th ee cases The chick et . of She t e-i mace m-
terfaes for She fine, medi m, ad coarse sim dation me
respectively ~ = 0 05, 0 025, md 0 0125 The d mt-
tions of the coarse md medi m resolution simulations
me t = 0 76 md t = 0 68, re pectively No pecial
trQmfient is used for She level-set f motion inside She
ship These donations correspond to about f ee quar-
ters of z ship leng h based on the present normalization
For these durations, the flow is steady near the bow md
still evolvmg near She stern he fine resolution simu-
idtion is still evolving, md it is not possible at f is time
to present complete results Mme complete results will
be provided at the mposi m md m the discussion sec-
tion of this paper I ) The ship is centered in She compu-
tatimurl domain with She same fixed sirJcage md h im as
used in She experiments in order to con truct the body
fm ce term , She hull is panel ize d us dry approximately
4000 panels
Coarse a d medi m resolution simulations have been
pe formed using the CLS formulation The coarse sim-
nidtion uses 256 x 64 x 64 g id pm · i. md She fine
resolution uses 512 x 128 x 128 g id points The
length, widdh, md height of She computational domain
are L = 2, W = 0 5, md 11 = 0 5, respectively The
water depth is d = 0 25 Th g id Pacing is constant
zlong all th ee cane i m axes ~ the next phi He of our
research, we will implement g id shetchmg, which will
all w g eater water depths to be simulated The d u t-
tions of the CLS simulations are t = 0 75 Unlike She
level-set results, th CLS re itsf ts extend the fiee-surfae
i terfae into She h al using She tech iq~xs o timed ear-
lier m our paper
The fre-smfae elevation was measured at DTMR
wasmmexpeotedlf~ctdow for fi e days of m mtenmce~m before
fi is ~ Per was due
using z whisk r probe Tw nty-one h traverse cuts were
pe formed near She bow, extendmg fiom ~ = 0 to
~ = 0 178 in dimensionless mits The whisker probe
measures the highest pomt of She free surface in re-
gi ms where here is wave t Gel i g, She whisker probe
m easures the t op of She be eaking wave Seventeen h ms-
verse cuts w re performed m th stern, extending fr m
= 1 01 to ~ = 1 22
Figures 3 md 4 compare measurements at She bow
md stern to the m me f ii tl predictions The b w mew
smements include profile md whisker-probe measure-
ments Comparisons to the b w data are performed at
four tations: ~ = 0 0444, ~ = 0 0622, ~ = 0 0800,
md ~ = 0 0978 The circular symbol denotes profile
measur merits The solid black lines denote the outline
of the hull md the whisker-probe measurements The
solid blue Ime is medi m CLS md the dashed blue Ime
is coarse CLS Th solid ff d Ime is medi m level-set
md the dashed red Ime is coarse level-set in general,
th CLS technique captures th mpid rise up the side
of the hull The level-set technique does less w 11 in
this regard ~ the outer-fiow region the CLS coarse re-
sults are slightly better th m the CLS fine res dts This
may be zth fluted to the shallow depth that is used in
th CLS The level-set results appear to converge bet-
tef in the outer-fi w r gi m, but She re mlts of fihe fine
simulation f re requi ed for confirmation
Figure 4 sh ws the enti e flow f mumd fihe ship for
th medium resolution level-set simulation The ste m
whisk r-probe mea smements are Merle id for fihe pUf-
poses of comp If if on Although the m mericaI results
are not sfatiorrtfy, the shape of fihe stem contours
sh w gee f a I a g cement with Dora tory measurements
How ver, the f mplitude of fihe m mericaZ remits f f e sig-
nifi mtly low r thm the measurements Note that fihe
stern is pa ftia lly d y m the m mericaZ simulations The
cull if f of fihe hull is visible m the m merica I simulations
because the If l-set fumcti m intersects fihe hull
6.2 Spray Sheet Results
The Na ier-Stokes eqmtriom in c mbination with f
If l f t formulation are used to so dy th breakup of
two-dimensiornl sheet of wate' The sheet is ZO = -mm
thick The length of fihe sheet is 24mm The top md
bosom of fihe sheet f f e bounded by f i' The initial
mem-velocity of the Water is f 0 = 3m/s The ini-
tiaI mms f rbuZent velocity of fihe Water is i = 1 2m/s
The fir is initially quiescent Based on th sheet fihick-
r ss To) md fihe me m velocity (f 0), the Rey olds m m-
ber is to = f 0Zo/q = 18,000 md the Weber m m-
ber is We = Of 2Zo/O = 730, when ~ is fihe km -
matic viscosity of WE ter, p is fihe WE ter density, ad o
is the f race tension The de sify md viscosity ratios
OCR for page 772
ate ~ = Pg/Pt = 0 0012 md g = ps/pr = 0 018,
which are appropriate for al -water interfaces his pa.
remoter regime rough y con esponds to experiments that
w re pe formed by Sarpkayc ad Merrill (1998), [14]
N mericcl convergence is established using 20489 md
40969 g id points Second-order ace Lacy m space is
established A fhird-order R mge-Kutta scheme is used
to integ ate The system of equations with respect to time
Mass is conserved to withm 0 25% th oughout The ff~-
ti e calcoktion
Figure 5 illushates the evolution of c two-
dimensiom~l spay sh et he black contour lines m-
dicate the mte fax betw en air md wate' he water
sheet is bo aided by air bodh et The top md the bott m
of the sh et he color contours denote the vo ti it
he -l w is turbulent within the water sheet md laminar
in the air Ike mea velocity md mms velocity profiles
me initially top-hat f motions he -I w is mm rig fr m
left to right he turbulent fluctuations in the water me
initially immersed bel w The top of the sheet md clove
thebottom of the sheet (see Fig 5: t = 0)
he turbulence m the water dfff ses ad interacts
with the mterfaes (see Fig 5: t = 2 5) he mitial
interaction is c roughening of The air water mterfae
A Thin bo mdary layer forms m The ail he bo mdary
layer is colored blue (negative) et the top of The sheet
md colored red positive) et The bottom of th sheet As
the mte 19c gets rougher md ligaments begin to form,
the air separates from The back of the ligaments he
bo mdary layer Thick m, md Hi is d egged along The
top md the bottoms of the sheet
Primary vo t :x shedding initially occurs behind The
ligaments (see I wer left of sheet m Fig 5: t = 5) As
the primary vo ti es are sh d, thei mre tcricns lead to
the fommation of secondary md tertiary vorticity (see
upper middle of sheet in Fig 5:t = 7 5) Vo tices
me periods ally shed from th back of ligaments (see
lower middle of sheet in Fig 5: t = 10) here is evi-
dence of vortex merging both m th air md m The water
(see upper led of Fig 5: t = 17 5) Aldhough theme is
sig iflc me -I w separation m the sir There is little or
no separation m The wate' he hugest ligaments me
fommedby eddies impingmg on The interface (see upper
left of Fig 5: t = 12 5) CCtvities fomm m regions whence
primary vo ti es me tapped Ibe inlets to the cavities
shed secondary vo ticity, which tends to make the cavi-
ties even Urger (see middle of sheet m Fig 5: t = 15)
At The inlets to The cavities, vortex phi s are formed Un-
der Their wn seff-induced velocities, The vortex pairs
move mto The cavities whence they din se
Note that d oplets do not actually form et The tips of
the ligaments be mse 2d -I ws me not Abject to The
same im tabilities es 3d -I ws he turbulent kinetic ff~-
ergy tend to conce Irate in The thicker portions of The
deformed prey sheet he flow withm the ligaments is
relatively benigm in cg cement with theory, The pres-
sme et The tips of The longest ligaments roughly scales
I ke P = (War) i, where r is th radius of curvature
of The tip
7 Conclusion
In this paper, w have outlined the key m mericcl cl-
gorithms for simulating free-smfae flows m cartesim
g ids usi g level-set md coupled level-set md vol me-
of-fluid techniques Preliminary m mericcl Results have
been show for ship waves md spay sheets he ship
_ He results indicate Nat cartesi m-g id methods are car
pable of resolving the flow aro md c ship if the g id
resolution is s fflcient Near the bow md rem, w
estimate that the g id spacing along ail f ee cartesim
axes should be ~ = 0 0005 abased on ship length) m
order to resolve breaking waves On c parallel com-
puter, it is possible to cpproah this level of g id res-
ol Lion, but adaptive g idding may also be hequi ed to
fully Resolve the entire flow aro md c ship [15] Alter-
m~tively, cartesim-g id methods could be mbedded in
more ccmemmmtl bo mdary-fltted methods to eeptme
eompl :x -l ws near The bow or rem he sprsy-sheet
results show that 3srtesim-g id methods are capable of
reselvmg the an ad water be mar y Dyer at reslisti3
R y elds n mbers
Aehmwledgmmt The -i st subhor is supported in
part by NSF D i -is i en of Mathemst ical So ience s mder
shard m mber DMS 9996349 The second subhor is
suppe ted by ONR mder 03ntrat m mber N000 14-97-
C-0345 D Edwin P. Reod is th prod em manager
The m merical simohtieus have been perfumed en The
T3E computer at the Naval O emeg aphid O flee us-
ing f mding provided by a D psrtment of Defense Ch~l-
lenge Project We ah very g steful to M Geor Le Immis,
D James Rettmm, md M A d w Tsl33tt for sssis-
ta e wish this pape'
References
[1] A S Almg en, J. B. Bell, P. Celelh, md
T. Msrthale' A 3srtesi m g id prot non medhod
for the no ompre s sib le enter e qusti ons in o m p lex
geometries SIAM J. Sei Camp t, 18(5):1289
1309, 1997
[2] J. B. Bell, P. Coke a, md H. M Glaz A
seoond-3rder prot non method for the moom-
pressible No ier-Stokes equations J. Camp t
P. a, 85:257 283, D comber 1989
OCR for page 773
[3] J. U. Bmckbill, D B. Kodhe, md C Zemah A
contimmm medhod for modelmg surfae tension
J. Comp t Phys, 100:335 353,1992
[4] Y. C Ch mg, T. Y Hou, B. Men im m, md S Osher
Euleri m ccptming medhods based on c level set
fomm ok t ion for mc ompre s sib le flui d i te faes J.
Comp t Phys, 124:449 464,1996
[5] P. Colelk,DT G'aves,D Modimo,EG Puck-
ett, ad M Sussmm An embedded b md-
ary/vol me of fluid method for fiee surfae flows
in ineg lar geometries ~ proceedings of 6he
3nd ASME/JSME joint ft~d engineenng confem
mce, n mber F DSM99-7108, Sm F'ancisco,
CA, 1999
[6] RG Cox The dynamics of the spreading of liq-
uids on c solid surfae part I viscous flow J.
Fluid Mech ,168:169 194,1986
[7] D G Dommermuth N mericcl Flow Arulysis
( fc) workmg pcpers Techmiccl ~eport, Science
Applications Interrutiom~l Corpomtion, 2000
[8] D G Dommermubh, G E mis, T. Luth, E A
Novikov, E Sch cgeter, md J. C Talcott N mer-
ical simulation of bow wa~s In pnoceedings of
the Tuzmh Saond Symp slum on Novol Hydro,
pages 508 521, Wcshi gton, D C ,1998
[9] L M Hocking md A D Rwxs The spreading of c
d opbyccpillaryation J. FhidMech, 121:425
442,1982
[10] H. Lcmb Hydrody mics Dowx Publications,
N wYork,1932
[11] B. J. L onard Bo mdedhigher-orderupwmdmul-
tidimensiom~l flmitewol me cor~ction-dfffusion
clgorif ms ~ WJ Ml kowycz md EM Spar-
row, editors, Advances in Numezicol Heot Tzans
fez, vol me 1, pages 1 58 Tcylor & Francis,1997
[12] CG NgmadEB D s mV O fhedynamicsof
liquid spreading m solid surfaes J. Fluid Mech,
209:191 226,1989
[13] S Osher md J. A Sedhia Fronts propagat-
ing wifh curvatme-depff~dent sp cd: Algorithms
based on hamiltonjaobi formulations J. Com
p t Phys, 79(1):12 49,1988
[14] T. SarpkaycadC Menill Sprcyfommationctfhe
flee smfa of liquid wall jets ~ proce dings of
the Tuzmh Saond Symp slum on Novol Hydro,
pages 796 808, Wcshi gton, D C ,1998
[15] M Sussmm, A Almgff~, J. Bell, P. Colelk,
L H well, md M Welcome A cdaptive level
set cpproah for incompressible two-phcse flows
J. Comp t Phys, 148:81 124,1999
[16] M Sussm madE G Puck tt Acoupled level set
md vol me of fluid medhod for computmg 3d md
axi mmetric incompressible two-phcse fl ws J.
Comp Phys acepted for publication
[17] M Sussmm, P. Smerekc, md SJ Oshr A
level set aproah for computmg solutions to m-
compressible two-phase flow J. Comp t Phys,
114:146 159,1994
[18] M Sus m m ad S Uto Computmg oil spread-
ing mdemeadh c sheet of ice Techmiccl R po t
CAM R po t 98-32, University of Cclffornic, Los
Ang les, July 1998
[19] HS Udayk mar,HCKm,W Shyy, mdRTra-
Son-Tcy Multiphcse dynamics m arbihary ge-
ometries on flxed cartesi m g ids J. Comp t
Phys, 137 2):366 405,1997
OCR for page 774
(a) x=0.0444
0.035
0.03
0.025
0.02
0.015
0.01
0.005
To
-0.005
0.035
0.03
0.025
0.02
0.015
0.01
0.005
To
-0.005
(b) x=0.0622
; ~ I_ _ >~ _ ~
0 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
(c) x=0.0800
(d) x=0.0978
~ __ _ At_ _
0 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Figure 3: Flow near bow.
OCR for page 775
O. :~6
.:: :~o ~ -
o 4~—
- 0.: ~—
-0. 50 -0. 25 0. OO ~0~ 25 O. 50 ~ 0. ~75 1. ED 1. 25 1~ 50~ 1. 75
x
:-~. 005 0. OO0 In:. 00~5 O. 010~ ~'015 O. O207
f ree aurfa.~e ale~ti:~n
Figure 4: Flow near stern.
OCR for page 776
-_:R f
F=~' ~ ~ I:
~ I^~4 ~
_ _
W_: ,.......
Figure 5: 2d spray sheet.
OCR for page 777
if ~4
-~
Figure 5: 2d spray sheet continued.
OCR for page 778
DISCUSSION
U. Bulgarelli
Instituto Nzzionale per St di ed E periep e di
A chitetturz Nzvale, Italy
In your algorithm do you have already adopted
the zdaptiv grid in He 3D geometry?
AUTHOR'S REPLY
We are m the process of dev loping z body-fitted
method The method will be described It the
next .-.~npp nun
DISCUSSION
K Hend ickson
Massachusetts institute of Tech olo :, USA
The mthors of d is paper show m zggressiv
use of z mmmerical method which has, to date
only, been used for smaller en m ermg
problems The fact that Hey are ah mpting it
for this type of problem says much about their
patiep e Ed ambition The blending of the
volume of fluid and lev I set methods is quite
creativ Ed shows promising re pits I believ
that the Cartesi m Grid Medhod is wonderfully
useful in fast m my of She gridding difficulties
hav been remov d Ed or Educed to panel
method that has been dealt with in detail in the
literature I wish th m both luck as they push
the method further
Quesdom
I At d is stage in dev lopment of the two
methods, it seems fast the coupled
lev l-set/volume of fluid technique (CLS) is
more zccurate/robust m t~eatmg the hull
boundary conditions mainly 1 e. mse it m
better defile where the hull lies in the
Cartesi m grid Hzv She mthors don my
mv stigation on She effects of the mollified
body force term used in the lev l-set LS)
technique? In using comparisons to She
water lip measm ement s as the benchmark,
She exact hull position would likely be z
critical point is it possible that the mollified
body force term is smoodhmg out the hull to
She extent that it is affecting the waterlip
recall ? Would less mollification or higher
resolution in the legion p or the body
produce better LS results? This c m almost
be i ferred fi om Figure 3 m the paper
2 The choice of friction coefficient in the
body force term (equation 19) seems to be
somewhat zrf irate Hz- e the mthors done
my type of parametric study on z r mge of
friction coefhcients Ed their effect on the
LS results?
3 Considerable effort has been mv sted in
the LS community to add ess rei ititli Zion,
which is also don in this paper The
reinitialization issue comes about he. mse z
Lzgr mgi m Thought process has been applied
to z E lerim method in most LS
formulations, the zdv ction of She lev l-set
fun non a, is perfommed using She v ID it
of She fluid This c mses He LS fun non to
lose its di tance fun tion property Ed requite
reinitiali Zion it is possible to construct z
v locity field Itch that the distance fun tion
remains on [1] Hzv the mthors considered
this type of LS fommubtion?
4 What is the computational cost comparison
betw en the CLS Ed LS methods at the
resolutions submitted in She p Her?
5 How do She hors feel CGM compare to
other less computttionallv expensiv
capabilities such as m FANS p 2D+T
methods?
6 What do the ~ hors consider to be She major
limitations of She CALL both CLS Ed LS, in
temms of then applicability to Maria
Hyd odypamics Ed Computational Ship
Hz d odypamics?
Referem es
I Adalste ins s on, D Ed Sethi m, J. A "The
Fast Conshuction of E tension Velocities in
L v I Set Methods," J. Comp Fhzit i, Vol.
I 45. 1999, pp 2 2
AUTHOR'S REPLY
I The coupled lev l-set/volume (CLS) of fluid
technique has z mme accurate treatment of the
hull boundary condition thm He lev l-set
method LS) Our te ts indicate fittt mollified
bod -force temms improv curse en e
2 The body-force term is as Urge as possible
without violating She Cour mt condition
3 The zdv outage of our remitialization
procedure is its accuracy, which c m be
gee ralized to my order Other procedmes,
OCR for page 779
such es the one proposed by the discusser, are
effective away fr m the interface
4 The CLS method is cutout twice es expensive
es the US method How ver, She computational
costs associated with both methods are less th m
ten percent of the Po isson so l- er
5 Irlre face tracking methods me capable of
ccpturmg physics that SIRENS md 2D+T will
never be capable of modeling Although
i te face tracki g methods are more
computationally expensive th m unRANS md
2D+T, f is will become less of m issue es
computers become faster Ten years from
today, inte face tracking will be the medhod of
choice for mod hng breckmg waves md the
near-field flow around real combat mts
6 The treatment of She hull boundary condition
is not accurate enough This issue is currently
being add essed by using c body-fitted grid with
c level-set treatment of the flee-su face elevation
in the near field of She hull in the outer-flow
region, the im r solution is mat bed to c
combirLttion of spectral methods md panel
methods This matching procedure reduces the
mmmber of grid pomts md the amount time that
is required to generate 3D grids
Representative terms from entire chapter:
body force
