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OCR for page 941
Analysis of Turbulence Free-Surface Flow around Hulls
in Shallow Water Channel by a Level-set Method
H H Chun, I R Park, S. K Lee Pusan National University, Korea)
ABSTRACT
In fLe presfft st dy on t rbuiffce free sufae
problem s m ha low wa er ch mnel, two fluids Reynods
ave~ged Navier-Stokes equaions a e solved by using
aFinite Volume Method, where 5 MPLEC agondEm is
used for velocity md pressure couplmg, md stmdad
k cturbulence model is inhoduced for modeling
Reynolds stresses A Level-set method is used for
capt ring the free-su fa e movement md the influence
of the t rbulence layer of fLe free su fa e is implicitly
considered For the vaidaion of the present numerica
scheme, fLe numffica results for Wigley a d Series 60
Ct =0 6 ships in deep water ae compaed with the
expenmffta results Computaions ae made for
vaious depth Froude numbers for fLe caculaions of
the halow water chmnel flow in the numerica
results, fLe presfft solutions show good agreemffts
with the experimenta results for the deep wmer ca e,
md for the ca e of fLe shalow water solutions with the
viscous effect, present numerica results how
rea onable physica phenomena in addition, it is
demonst~ted fha fLe level-set medhod cm trea the
free su fa e flows a ound hulls with a rea onable
a cu~cy togedher wifh a simple numerica procedure
INTRODUCTION
Shalow water ch mnel flow nea fLe critica depth
Froude number Fh(=U/~/~)=I is m unsteady,
nonlmea phenomenon md ha peculia flow
chamctenstics, where h is watff depfh, U is the ship
speed md g is fLe gravitaiona a cele~tion At this
critica speed, a ship gene~tes tw -dimensiona waves
propagaing m fiont of fLe ship, which ae fatff thm
the ship speed md show the unsteady flow paten~s
These waves a e named solitons or solitay waves The
influences of fLe ch mnel wal md shalow wmer c mse
the increa e of the resist mce md sinkage of fLe ship a
nea fLe critica peed
In the experimenta mvestigaions, Thew md
L mdweber (1935), Helm (1940), Graff et a (1964) md
E tekin (1964) obse~ved these unsteady md nonlmea
waves in shalow wmer towmg tmks Especialy,
E tekm (1964) carTied out a series of expenmffts in
which cetam paameters such a wmer depth, hip
d ad md ch mnel width were ch mged
For fLe inviscid flow problem s, Bai md Kim (1969)
used a Fmite Element Method for solving a nonlinea
fiee-su fa e flow for a hip moving in reshi ted
shalow watff tmk in cae of the numerica w rks,
Choi md Mel (1969) used Kadomtsev-Pehiahivili
equaions md E tekin & Qim (1969) md 3img (1996)
used Boussmesq equaions for solving fLe nonlmea
shalow wmer waves Kim md Lee (1996) investigaed
these phenomenabaed on Eulerequaions
For fLe viscous solutions in shalow water cha nel,
Bfft~m md Isha awa (1997) used the hybad apmoa h
computing fust squa a d potentia flow wifh fiee-
su fa e caculaion a d then fLe viscous flow wifLout
fiee-su fa e effect a fLe subcatica depth Froude
numbe' in fLeir numerica results, fLe pressure on the
chmnel bottom md hull su fa e ae compaed with
experimenta results, where it is explained tha the
dis~repmcies ae cmsed by disregad of the
defommaion of fLe free-su fa e
b the present wodk, the viscous md fiee-su fa e
effects ae considered m fLe caculaion of fLe shalow
wmer chmnel flow a fLe catica md super-critica
depth Froude number speeds For fLe a aysis of
turbulence flow, two fluids Reynols ave~ged Navie~
Stokes equDns me solved by usmg a Finite Volume
Medhod, where stmdffd k cturbulence model is
usedformodeling Reynolds stresses
For fLe fi ee su fa e h emment, tw main mpproffhes
(front ha kmg & fiont cffptunng methods) have been
used in fiont cffptunng methods, fLe level-set schffme
hff been only recfffly used in free su fa e probiffms
(Vogt 1998 Domme muth et a 1998 Bet et a 1996
md Pffk & Chun 1999 (a), (b)) The level-set method
is a numerica technique which c m follow fLe
evolution of mte fa es These mte fa es Cff develop
shffp comers, bremk mpfft, md me ge togedhe' The
level-set method hff a wide rffge of mpplicDns,
including problems in fluid mechmics, combustion,
m mufa tunng of computff chips, computer mimDn,
image processing, struct re of snowflmkes, ffd fLe
shffpe of SOff bubbles Especialy for mffy complex
fiee sufae problems (eg bremkmg wave, pray
OCR for page 942
A=
o
p(oul
P(o)U2
p(d )U3
p(tt Ik
p(tt.k
Front F x =
p (d )g, pl
pot Ig2 Pj
p(tt Ig3 pk
Gt PI
(C,Gt C p(o,k I
P(Ou o
p(d)UU, U,t (d )(Vu~ )
p(d )uu2 U,t (d )(Vu2 )
p(O)UU3 U,t (d)(VU3)
p(d )Uk ~ (Vk)
p (d )U~ at (d ) (Vet)
(3)
where A: file pressure, k: t rbulent kinetic energy,
c: turbulent dissipation rate, uqq = u+ a. :,
turbulent eddy viscosity md Gt: rate of production of
turbulence kmetic ene gy is defined as
Gt = U ''(_ + aU2 + aU3 , _ (dU2 + a,.,
ax By as ax By
(au3 + aU2 )2 + (are + au3)2) (4)
By ax d_ ax
Turbulent constmts of file stmdard k
turbulence model is defined in Table I
Table I Turbulence constants of the
k ~ .,, balance model
as
phenomena, slamming problem, water-exit problem
md bubble problem) in file naval hyd dynamics, this
method is aprovocative approach md could be used as
a robust numerical scheme in file present wodk, d is
level-set medhod is introduced for cant ring the free-
su face movement md implicitly considermg the
influence of turbulence layer of the fi ee su face
In o der to validate file present numerical scheme,
the expert emol results of Wigley md Series 60 ship at
deep water condition are compared wish file present
numerical results in file numerical results of the
shallow watts flow, the wave patterns, pressure
di tributions on file hull md the fiction md pressure
resistmces computed at different flow conditions are
compared with each of her End pe tmert discussions are
included Fr m these numerical results, file validation
of file level-set method c m be also checked
MATHEMATICAL FORMULATION
Governing Equations
In the three-dimension problem, the general integral
fomms of file time-weraged conservation equations of
mass, two-fluid mcompressible Navier-Stokes
equations md turbulent kmetic energy dissipation rate
c r be Isomer
111 J3t (p(d a) din
+|| (p(t~luq R(d)Vq)o is=lll SgdQ
(1)
sphere u is file velocity of fluid, q is my
conservative qu mtity, R is the associated diffusive
coefficient md So is the volumeh ic source te m of q
In the present Rests, the stmdard k
turbulence model is used for file turbulence flow
Equation (1) c m be w men in file vectori ed to m
||| a (U)dQ+|| (Front Fd~r)odS
= iii B dQ (2)
where sectors U. F, v F x End B are defined
as
st mdard
C C, C2 ~ ac X
009 144 192 10 X 041
(C, C2)~
LEVEL-SET FORMULATION
As semi in Fig 1, file immiscible md incompressible
tw -phase fluids are described by their densities ( A,
P2 ) md viscosities ( U . u2 ), where these physical
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Representative terms from entire chapter:
shallow water
reinitialization procedure, sign(dO)=1 if do >0'
sign(dO) =—I if do < 0, and sign(O) = 0.
Second order END scheme is used for spatial
derivative |Vd?| in equation (14) and implicit Euler |V¢|—I
scheme is used for the time integration.
The idea of this equation is that a steady-state
solution will be a signed distance function which has
|Va7| = I near the interface with the same zero-level-
set as the initial function do (x) . This method
generally works well when level-set gradients are
neither too flat nor too steep near the interface. The
value al(x,t)propagates with speed +1 along the
characteristics that are normal to the interface and
converges quickly in a neighborhood of the interface
for small time t. The fact that a'O(x) is already a
good guess for the distance function is effectively used
for obtaining the convergence solution after a small
time t. Because the interface moves a little, the
previous information can be effectively used to update c(~) =
the new level-set distribution by an iterative scheme.
The width of the neighborhood around the interface
can be defined Mh, where OK > 0 and it is
sufficient to solve equation (14) up to time Ash.
Since the characteristics are propagating away from the
interface with speed of unity, the appropriate time step
according to the CFT (Courant Friedriches Lewy)
Ah
condition is At =
2
Local Level-Set Method
If the level-set function around the interface is an
exact signed distance function, the magnitude of the
level-set function gradient at the interface must be
unity, namely,
(17)
The localization makes it only necessary to perform
the evolution and reinitialization of the interface within
a narrow region around the front.
Let O < ~ < The two constants that will be
determined according to the grid size. Around the
interface FO, a tube with width r can be defined by
To ={x~ Ret :|4pO(x)|
remitializ tion procedure, sign(dO) = I if do > 0,
sign(dO)= I if do <0. md sign(OI=0
Second order ENO scheme is used for p tial
deriv h -e |Vd| m equ tion (14) md implicit Euler
scheme is used for file time mtegmlion
The idea of this equ tion is th t a ste dy-st te
solution will be a signed distmce f nction which has
|Vd| = I near the inte face wish file same zero-level-
set as the initial f nction do(x) This method
generally w rks well when level-set gradients are
neither too fl t nor too steep near the mte face The
value d(x,t)propagtes wish speed +1 along the
chamctenstics th t are no mat to the inte face md
conve ges quickly m a ne~ghbo hood of file mte face
for mall time t The fact th t do(x) is already a
good guess for file did mce function is effectively used
for obtaming file conve gence solution fter a small
time t Becmse the mteface moves a little, the
previous mfomm tion c m be effectively used to upd te
the new level-set distribution by m net tune scheme
The width of file neighborhood a ound file mte face
c m be defined tush, where t > 0 md it is
sufficient to solve equ tion (14) up to time tush
Since file chamcteri tics a e propag ting away from the
inte face with speed of unity, the appropn te time tep
according to the CFL(Courmt Fnediches Lewy)
Local Level-Set Method
Acco ding to file locality prope ty of the level-set
method, it is mffcieDt to calcul te file level-set
function only m a small naTow b md around its z~so-
level-set for the reinitializ tion procedure By doing so,
in file case of tw -dimensional comput non let
IV X IV be the number of gad point, then
comput tional expense reduces from O(iV ) to
O(N) (Peng et al 1999) A PDE-based local level-set
method We developed by Peng et al (1999) The local
level-set method described m fLeir paper c m be
summari ed as follows
The level-set function must say well behaved
except for isol ted pomts for numerical accmuc!:
O
c on stru cte d fi om ~l ( x )
¢~(x)=d~(x) for|d~(x)|
md diffusive fluxes at the cell-face cemer md each
flux c m be appr ximated by using file midpomt r le as
follow:
Mass flux:
he = || p (d )U o dS = [ p (d To n ], 5~ (34)
Convective flux: /e qe ( )
Diff sive flux: [rg(d)Vqo n]eSe (36)
The hybrid scheme is used to appr ximate
convective fluxes
For file approximations of the diff sive fluxes, the
gradient ve tor at the cell face c m be calculated fir m
the gradients at file cell comers appr ximated by using
midpoint r le based on the G mss theor m:
Eli (Vq) dQ || q t o dS ~q,S,
(vq)r= AQ ~
,(c=e,w,n,s, ) (37)
AQ AQ
But this approach may c mse oscillate y solutions
(Per iger md Peric 1996) Muzafenja (1994) noted
that the followmg implicit expression is to hit for
approximations of diffusive fluxes when the
orthogonal regular gad sy tem is used
(V = qr In
(3g)
As seen m Fig 2, when file gad is in egular, file line
connecting points P md E does not pass through the
cell face center e For the case of a non-orhogonal
md inegular grid system, Muzaferija (1994) sugge ted
m effective scheme, 'differed connection medhod', to
prevent the oscillate y solutions:
(vq)e ° ne =
a,' Lo I d!' (aaq;)e]°t
(do = qr no +[qr qr' (Vq)e ot;]°t
an Ll e Lfrf
=qr He +[qr He
L r Llfrf
(vq)r ° (rr rr) (Vq)~ 0 (rt. rig)
~ e (Vq)r + (I ie.)(Vq)t) o z; ] t
when i;: file unit vector m file ~ -dire tion md
overbar denotes mtepolation trom neighbor nodal
values
qr =qt+(Vq)to(r~ rl I (40)
Irr rrl
r = re [ e _ ]ne
r =re [(re r )°nr]ite
if ne=nr
(41)
(42)
(43)
f file ire conne ting points P md E is orthogonal
to file cell face, file oldbracket -. - ~ is ZffO md usual
the cenh al differ nce approxim ation of file derivative is
recovered The explicit -. - ~ at the cell face is used to
consider the influence of the cross-denvative On a
non-orhogonal inegular grid wstem, values of fluid
prope ties at file cell face center c m be approximated
qe qr e qr( e) qr e qI( e)
+(Vq)ro(rr. rZ)~,+ Vq)ro(rr~ r~)(l be)
(44)
The implicit Euler scheme is used for the time
integration as follows:
if a' f (q (t)'t) ~ d' = 0
~ q~+~=q~+At[f(~+~'q~+~)] (45)
The disco ti ed mea algebraic equation system
cm be solved by using file SIP (Stone 1966) solve'
SlMPLEC(Semi-lmplicit Method for Pressure-Linked
Equation Consi /en ff algorithm is used for pa ssur -
velocity coupling on cell-center d grids (vm Doonmal
mdRathby 1964)
NUMERICAL RESULTS
In order to check the accuracy md Ableness of the
preset numerical scheme, file teady wwes for tw
ships (Wigley md series 60 ships) in deep watts are
calculated md compared with expenmemsl results
published in addition, the numerical results published
for these tw hips by other CF tools are widely
available md file accuracy md ewe tiv:~ess of the
present numerical results c m be mdirectly compared
The computational conditions for Wigley ship md
Series 60~C =061 ship are as follows: body length
L =2 5, FN = 0 289, RN =3 277xlO6,
80x26x40CVs md I l Ox40x60CVs for
Wigley ship FN = 0 316, RN = 4 0 X 10 md
I l Ox40x58CVs for Series 60 hip, where H-H
type gad is employed for the present calculation
Fig 3 show file time hi to y of file side wave
profiles along Wigley hull at FN = 0 289,
RN = 3 277 X 10 with time intervals of 0 7s from
10 5s to 18 9s md file experimental results
(Kajitmi et al 1963) are included to check the
accuracy Aldhough there are some discrepmcies to
notice around the bow, it c m be seen Hat the
calculated results have consistency wish doff erent times,
which me us the converged solution wish a high
stableness in addition, by comparing the odher
numerical results publi bed for flus wave profile, it c m
be regarded that the accuracy of the present numerical
results is good
Fig 4 show file contour of the waves generated by
the Wigley ship for tw grid densities it c m be also
seen fi om file comparison of the other numerical results
published Hat the present results give a reasonable
accuracy
Fig 5 show file level-set function contours
conve ged along file hull of Wigley ship at
FN = 0 289 The dishibution of file level-set
function show the unifo m Slickness along the
inte face sphere the thick solid ire is zero level-set,
namely, file h ee su face
Fig 6 shows the wave profiles calculated along the
Series 60 ship hull togedher wish the experimental
results at FN = 0 316 md RN = 4 0 x I O A
good agreement between file tw results cm be seen
The pressure dishibution on the hull su face is also
seen m Fig 6, showing a smoodh pressure variation on
the hull so face
In Fig 7, file wake dishibutions calculated for the
Series 60 with file same condition as in Fig 6 at the
tw plmes (A P is x/l=10) are how togedher
with file measured results Toda et al 1991) it is rather
disappointing to see that present numerical results do
not agree well with file experimental results However,
it cm be viewed fiom file comparison wish odher
numerical results published Hat almost file similar
level of numerical accuracy is obtained for the present
turbulent model used b recent published paper, it is
recommended that more cone ted md appropriate
turbulence models should be used for a good prediction
of file wake dishibution rear file ship hull md gad
density dependence of file solution is not so critical
(Hino 1994)
Fig 6 show the t rbulent kinetic energy md eddy
viscosity di tributions around the hull of Series 60 hip
at FN = 0 316 The fi ee so face turbulence layers in
the an md file water region c m be seen The Shot new
of file turbulence layer developed around the fore past
of file hull boundary on file free su face f ther
develops End becomes thicker as it moves backwards
Although file numerical solutions computed by file
present scheme near the free su face might not be so
exact, the calculated t bnlent properties show
reasonable physical characteristics
For file shallow ch mnel watts calculations, file
Wigley hull used previously is taken in file numerical
procedure for hallow water ch mnel problems,
blockage coefficient 5 = Al /(2wh) = 0 021, a
ratio of water depth to the hull d ft It / T = 1 598
md a ratio of chamel widdh to the hull length
w/L=20are used, where Aois file ~ross-section
area of the hull at mid hip at a given daft Fig I show
the definition sketch for file shallow water chmnel
problem For file computational cases of Fh = I md
Fh =1 5, RN =3 6X10 md RN =5 0xlO
are used, respe tively The computational chmnel
length x / L = 10, where x / L = 5 head of the hull
md x / L = 4 behind the hull in each computation,
210x44x46CVsis used md H-H type grid is
employed for the present calculation Computations a e
stated at full peed without flow acceleration No slip
flow condition is used at the hull boundary md the
sidewall of file chmnel, md the symmet y boundary
condition is inhoduced at the bottom md also at the
center pure of the ch mnel
The numerical results are show in Fig 9 Fig 9 (1)-
(a) shows file steady wave contours at FN = 0 316,
RN = 3 6 x l O for deep watts which is equal to file
critical speed Fh =I for file shallow chmnel The
unsteady wave contours at this critical speed with
different times for file hallow water chmnel se
plotted in Fig 9 (N-(al-(cl Unlike file te tdy case, the
unste tdy wave p latent wish a soliton Ph p tg ding ahe Id
of file ship cm be observed it is knowt thy this
phenomenon is c msed by the effects of blockage md
shalow wme' md file various vanes of parameters,
h/T, w/L mdblockage coefficient, may crease a
little different flow phenomena
In addition, the results for Fh = 1 5,
RN = 5 0 X I O a tw different times a e included in
Fig 9 (i~(ai-(b) it is generous know fha a soliton
develops a d propaga es to wa d of file bow of file hull
a the critical speed However, a seen m Fig 9 (3)-
(a)~(b), file wave patenh a the supercritica speed
ret as to file teddy wwe patem wish only the
divergent wave system This wwe patem is simile to
hypersonic flow, md file mgle of file divergent wave is
about + sin (I /Fh) = 42
In Fig 10, wave profiles caculaed Song file hull
su fa e a FN = 0 316 in the deep wme' a the
critical speed Fh = I md a file supffscritica speed
Fh = 1 5 m the shalow water ale how it cm be
seen fha the difference of file wave heights md
pmerns between the results for file deep waler md
shalow charnel water is somehow considerable By
compa ing the wave profiles for file deep water md the
shalow charnel of Fh =I whose tw speeds are
equivalent, file iagff wave nea the bow md a deep
trough nea file steno for the clitics peed ca e, which
is of course cat sed by the big pressure ch mges due to
the shalow water md ch mnel effects a seen laser in
Fig l2, crepes a large him by .-. -I
For file three computations ca es, the pressure md
faction resistmces acting on the hull are plotted in
Fig 11 it cm be seen that the shalow water chat nel
effect mcrea es file pressure resist mce md the
magnitude of the resistmce ha file ma imum value
nea file critical ship speed in the shalow waler
ch mnel
In Fig 12, the pressure dishibutions on file hull
su fa e me plottedforthe fEree ca es Smce file ewe ts
of file bottom md side ml is of file ha low ch mnel a e
considerable, the magnitude of file pressure difference
Song file hull is higher for file critical speed ca e th m
that for the deep water ca e When a ship adh-arces in
the shalowwmer charnel a ahigh speed, file pressure
charges on the ship hull cased by file ewe t of the
shalow charnel crepe a large bow wwe with a big
trough nea the stem a seen above md a cordmgly,
case file severe sinkage md him of the ship, aso
resulting a la ge resist mce increa e
In Fig 13, the pressure di tributions on file bottom,
the cent epl ~ e md sidewal of the charnel a
Fh =lae show Becmse of the wave propagation
ahead of the hull, a high pressure dish bum on s over the
upstream region cat be seen
b Fig 14, the velocity distributions md a la
velocity contours a x /L = 0 75 a d x/L = I O
a e show for file fEree computations ca es Smce the
wave heights Song the hull su fat e a d file ma mt des
of the fluid velocities for the three ca es a e not a I file
same, file a la velocity contours show somehow
different shmes Becmse of the effects of the bottom
of file ch mnel, the a la velocity contour a the hull
bottom a e widens for file ch mnel flow ca e th m fha for
the deep waler ca e
CONCLUSIONS
The level-set apron h to solve the t rbulence fiee
su fat e flow a und hull m deep wmer md hallow
waler charnel ha been developed The advmtages of
the level-set medhod make stable computation
procedure, eaw pm ~ [mns~g md aso gives
rep onably a curate solutions for the present viscous
hee-su fa e flow it seems fha the level of the
numenca a curd y obtained by the present medhod is
simile to fha of of her methods used with the same
turbulent model a file present one Numerica results
of the fiee-su fat e flow Sound a hull in shalowwmer
ch mnel with file viscous effects show rep onable
physical phenomena But in the f ture, it is necessary
to compare numenca results wish expert ems for
validation md Only out more wstemaic study with
vaiaions of parameters, h/T, w/L md blockage
coefficient
FtEFEFtENCE
Bai, K J. md Kim, J. W. "Numerical Computations for
a Nonlmea Free Su fat e Flow Problem," Proceedings
of the 5~ Intemaiona Conference on Numffica Shin
Hvd dynamics, pp 403-419, 19S9
Bet, F. Hazel, D md Shanna. S. Iiumffical
Simulation of Ship Flow by a Medhod of ArtiEcia
Compressibility", Proceedings of the Twenty-Second
Svmnosmm on Naval Hvd dynamics, Wahmgton,
pp 173-lS2, 1998
Choi, H S. md Mel, C C, "Wave Resista ce md Squa
of a Nader Ship Moving Ned the Cntica Speed in
Restricted Wrier" Proceedings of the 5' Intemaional
Conference on Numenca Shin Hvd dynamics,
pp 439-454, 1989
E tekm, R. C, "Solution Generation by Moving
Dish rbmces in Shallow Wow: Theo !, Computation
md Experiment," Ph. D Thesis, University of
Califonua, Berkeley, 1984
E tekin, R. C md Qi m, Z. M, "Numerical Gnd
genetmmn md Up me m Waves for Ships Moving,"
Proceedings of the 5~ Intemstionsl Conference on
i, pp 421-437, 1989
Dommemmuhh, D, b nis, G. Luth, T. Novikov, E,
Schlagete' E md Talcoh, J. "Numerical Simulation
of Bow Waves," Proceedings of the Twenty-second
Symposium on Nwsl Hydrodynamics, Washmgton,
D C, pp 159-172, 1998
Ferziger, J. H md Penc, M, Computstionsl Methods
for Fed d Dyn am ics, Spy Go -V erlag, Berlin, 1996
G off, W. K acht, A md Wemblum, G. "Some
Extensions of D W. Taylor's Stmdard Series,"
Trsnwctions of Society of Nwsl Architects and
Msnne Engineers, Vol. 72, 1964, pp 374-401
Hasten, A, md Engquist, B. Oshe' S. md
Chakravarthy, S. "Unifo mly High-Order Accm lie
Essentially Nonosci IF u y Schemes, 111," Joumal of
Computstionsl Physics, Vol. 71, pp 231-303, 1987
Helm, K, "Effects of Chmnel Depfh md Width on
Ship Resi tmce," Hyd odynamische Problem des
Schiffsmbiebs, Pat 2, ed G. Kempf, Verlag
Oldenburg, Munich md Berlin, 1940, pp 144-171
Hino, T. "A Study of Grid Dependence in Navie~
Stokes Solutions for Free Su face Flows a ound a Ship
Hull", .loumal of The Society of Nsval Architecs of
L~, Vol. 176, 1994
Ji m, T. "b vestigation of Waves Gen~sated by Ships in
Shallow Wat~s," Proceedings of the 77~ Symposium
on Nsval Hydrodynsmics, Washmgton, pp 601-612,
1999
Kajitmi, H. Miyata, H. Pkehata, M, Tmaka, H.
Namimatsu, M, md Ogiwma, S. "The Summary of
the Cooperative Experiment on Wigley Paraholic
Model b Japm," Proceedings of The 7~ DTNSTDC
Workshop on Ship Wsve-Resistance Computstions,
Bethesda, Marylmd, U S. A, 1983
Kim, S. Y. md Lee, Y. G. "A Sh dy on Upstream
Waves for m A bih a~y Hull Shape m Restacted Water
Ch mnel," Proceedings of the Anm~sl Auh~mn Meeting
SNAK, b chon, 13-14, November, pp 107-112, 1998
(m Kore m)
Muzaferija, S. "Adaptive Finite Volume Medhod for
Flow Predictions Usmg Unstmctured Meshes md
Multigad Approack," Ph. D Disse tation, University of
London, 1994
O her, S. md Sethim, JA, "Fronts Propagatmg with
Cmvature-Dependent Speed: AlgorifEms Based on
Hamilton-Jacobi Fommulations," Joumal of
Comput~ Vol. 79, pp 12-49, 19SS
Park, l R. md Chun H H. "A alysis of Flow around a
Rigid Body in Wate~Ent y & Exit Problems," Jonn~l
of the Society of Nsval Architects of Kores Vol. 36,
No 4, pp 37 47, 1999 (a), (in Kore m)
Park, I R. md Chun H H. "A Sh dy on the Level-Set
Scheme for the A alysis of fLe Free Su face Flow by a
Finite Volume Medhod," Joumal of the Society of
Nsval Architects of Kores. Vol. 36, No 2, pp 40-49,
1999 (b), (in Kore m)
Peng, D P. Osher, S. Zhao, H K md K mg M G. "A
PDE-Based Fast Local Level Set Method," Joumal of
Comput~Vol 155,pp 410-43S,1999
Stone, H L, "Iterative Solution of bmplicit
Approximations of Multidimensional Pa~tial
Diff
110X40xi0CVs~ ! ~
h
T
~ I: local level-set tube
_ _
air ~ ~ O Q1 (~1, Pi)
water ~ ~ O Q2 (~2, P2)
Fig. 1 Definition sketch (shallow water channel)
C an face
Cell center plane { e-direction }
- · \ In
e
—I_
EM
Fig. 2 A non-orthogonal & irregular grid
`1 `14 ~
0.04
1.~ ~ ~
3 3.~
Fig.3 Wave profiles calculated along Wigley hull; in
case of the calculation of 1 10 X 40 X 60CVs, history
of wave profiles at intervals of 0.7s from 10.5s to
18.9s shown (FN = 0.289, RN = 3.277xlO,
F.P.=1.5 & A.P.=3.5)
Fig.4 Wave contours computed for Wigley ship with
two grid densities
(FN = 0.289, RN = 3.277 x 10 )
F.P.
A P.
Zero level-set: free surface
Fig.5 Converged level-set contours along Wigley ship
hull, where the range of levels is from- 0.04 to
0 04 (FN = 0.289, RN = 3.277xlO )
~ Kalitani et al 1983
—cat. s~}~2~x4o cats
ton. 1 1 {~4ox~o con
(~.~< ~ <18.~)
~ ~ it\
o\ ~
n no
0.06
~ Toda et al 1991
Cal. llUx40~S CYs
~ 4
Fig.6 Wave profiles and pressure distributions along
Series 60 ship hull (FN = 0.316, RN = 4.0X10
F.P.=2 & A.P.=4 )
Fig. 6 (continued)
| EXPERIMENT|
| EXPERIMENT |
j~lt~ttt~t
Pitt
it t t
XIL=1.0
1 1 ~ I I ~
XIL=1.2 1I]1IJIJ// ~
,,
1 1 J J / / ~
1~\ ~ 1 ~ 1 J /
/, , ~
Fig.7 Axial velocity distributions and contours at x/L = 1.0 (A.P.) & 1.2: left side for experimental, right side
for numerical results ( FN = 0 3 1 6 ~ RN = 4 0 x 1 o6 )
Turbulent hi
etic energy
Wit A.r
Eddy viscosity
A.P.
Fig8 Turbulent ki etic em & eddy viscosity distributions around hull of S
(1) Deep water EN—0.316 RN = 3 6XIO6
1 \ ~ ~
(2)-(a) FL=l.O, FN-0.316, RN-3.6XIOS
Fig 9 Calmiated wave pattems of Wigley hull (1) atFN = 0 316 i file deep wate' (2) at the critical peed
Fh = I and (3) the supercatical speed Fh = I
St =0021)
with h/T=1 598 and w/L=20 (blockage coefficient
1.50E-02
1.25E-02
1.00E-02
7.50E-03
VP
>~w water Ph=l 5 ~
:
shallow water Fh ALP. |
. .
Fig.10 Calculated wave profiles along Wigley hull (1)
at FN = 0.316 in the deep water, (2) at the critical
speed Fh = 1 and (3) the supercritical speed
Fh = 1.5 for hlT = 1.598 and w/L = 2.0
(blockage coefficient Sb = 0.021 )
Cp ( Fh=l) At=0.0045
,
, ,
~ it,
_ ~ ~ ~ ~ ~I
,, Al, U
,, ,,'
, ,
_ , ~
-
~ ,' Cp ( Fh=1.5) At=0.0025
Ill ~~
~ ant. no
0.00E+OO'
- rev ~N=0~316) At=0.00425
1 1 1 1 1 1 1 1 1 1 1
2500 5000 7500
iteration
Fig. 1 1 History of the resistance of Wigley hull at
FN = 0.316 in the deep water, and at Fh = 1 and
Fh = 1.5 in the shallow water channel with
h/T=1.598 and w/L=2.0
(blockage coefficient Sb = 0.021 )
Deep water FN = 0.316
1 4 7 10 13 16 19 22 25
-103.3 -61.9 -20.5 21.0 62.4 103.8 145.2 186.6 228.0
IF.P A.P|
/ ~ ma\
Shallow water Fh = 1
1 4 7 10 13 16 19
,/ / ~
22 25
-270.4 -186.1 -101.8 -I 7.5 66.8 151.1 235.4 319.8 404.1
F.P.
Shallow water Fh = 1.5
1 4 7 10 13 16 19 22 25
-254.2 -173.4 -92.6 -11.8 69.0 149.8 230.6 311.4 392.2
Fig.12 Calculated pressure distributions on the hull surface at FN = 0.316 in the deep water (top),
atFh=ltmiddle) and Fh=1.5(bottom) in shallow water channel with hlT=1.598 and w/L=2.0
(blockage coefficient Sb = 0.021 )
3925
296.0
247.8
996
t21.3
-89 4
-185 8
-282.1
Shallow channel Fh = I, RN = 3.6~06
hlT= 1.598
w/~=2.0
Sb= 0.021
Fig.13 Calculated pressure distributions on the bottom, sidewall and center plane of shallow-water channel
atFh = 1, h I T = 1.598 and w/L = 2.0 (blockage coefficient Sb = 0.021
~ Shallow water Fh - 1 |
-
Shallow waterFh = 1.4
~ ~ ~ ~ ~ ~ Add_
Deep water FN 0.3161
| Shallow water Fh = :
. ,
,
.
_=
.. _ ~
Deep water FN = 0.316 .
(a) x/L=0.75
(b) x/L=1.0
Fig.14 Calculated velocity distributions and axial velocity contours at x/L = 0.75 and x/L = 1.0 at
FN = 0.316 in the deep water, atFh = 1 and Fh = 1.5 in the shallow water channel with hlT = 1.598 and
w/L = 2.0 (blockage coefficient Sb = 0.021 )
DISCUSSION
T. Jiaug
Versuchsausta t f r Binuenschiff au
e.V., Gem any
First, I wou d ike to congratu ate the
authors to attack this k ad of non inear
and unsteady problems using au
unsteady RANS, a though is can a so be
effectively approximated by using
shallow-water wave theory. For
instauc e, K- P- e quati on on B oussine sq
equations. The most benefit by using au
unsteady RANS cou d be the better
approximation of the wake waves near a
transom stem in genera and of the bow
waves at high supercntica speed, since
the bow waves are very sensitive to the
bow shape. Tur ing now to my
queshon. Is we look at your t me history
of the resistance, it does not seem to be a
closed periodic asymptotic solution.
However, both our expemmouta resu Is
as well as numenca resu ts have shown
these closed periodic procedures.
AUTHOR'S REPLY
We wou d ike to thank Dr. Jiaug for
nice and useful comments.
In this preseutahon of our work, the
ca cu ated hme history of the resistance
at the cnuca speed does not show a
periodic behavior. This can be expla Red
by two main reasons:
The f rst reason is that the pressure
correction equation cou d not be
ca cu ated accurately in the case of the
unsteady cutica speed flow condition
because of reducing the t me consume.
The second reasonis that our numenca
resu t is that when the second soliton
wave just propagates ahead of the bow
of a ship after one soliton wave
propagates.
Therefore, if more accurate ca cu anon
of the pressure cc era -con equation is
conducted and resu 15 of more advanced
t me steps are obtained, our numenca
solution at the cnuca speed may show a
accurate and closed periodic asymptotic
behavior.
DISCUSSION
N. Stuntz
University Duisburg, Gem any
In the simu anon the ship model is held
f xed du ing the cc mplllahon. In the
case of the ship flow in rest icted wastes
in the trauscntica range this might have
considerable influence on wave patterns
and resistance. Have the authors
brought about this by hav ug in m ad,
that the mesh is a so fixed during
computahon using the level-set method?
AUTHOR'S REPLY
We wou d like to a so the k Mr. Stuntz
for the comments on our research
As you referred to your discussion,
different conditions of a ship in the
rest icted she low water cha gel might
cause different flow patterns (wave
height and resistance, etc.)
However, it is not easy to simu ate the
ship at every iustaut when the ship
changes in trim and si kage at the
cnuca speed due to too much t me
consumpt on.
O r resu ts in every flow conditions are
computed with f xed g id system
More accurate solution can be obtained
by using g id adaptation method.
