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OCR for page 805
Computation of Nonlinear Turbulent Free
Surface Flows Using the Parallel Uncle Code
M. Beddhu, R. Pa kajakshan, M. Y. Jiang, M. Remotigue, C. Sheng,
L. Taylor, W. Briley, D. Whitfield (Mississippi State University, USA )
ABSTRACT
A m mericcl cpproah is presented m this work
suitable for She computation of nonlinear free surfae
flows over complex geometries such es ship hulls m c
fc t, reliable ad robust meaner The go coning equa-
tions solved are the incompressible Rey olds Averaged
Navier Stokes tANS) equations coupled with the free
surfae kinematic condition Ed c two equation tmbu-
lence model Simple no nommcl g cdient dynamic
bo mdary conditions are used et the free surfae The
governing equations are cast with respect to m msteady
(nor~merticl) general curvilmear coordi me >! tem
The mmericcl reproach uses She mod tea artfficicl
compressibility formulation The governing equations
are discretized using c to nite vol me approach where the
m mericcl flw.es et cell interfaces are obtained using
Roe's inviscid flux averages coupled wish m L er's
MUSCL formulation for higher order flux extmpolc-
tion Viscous flw.es are avenged using central dfffer-
encmg Time is discretized implicitly using She first or-
der Euler backward differencing The resulting
non I near algebraic eq onions are solved using the dis-
creti:D:d Newton relaxation AND) cpproah with sym -
mehiccl Gauss Seidel weeps To speedup the solution
process c parallel implementation of the m mericcl cl-
gorithm duct uses h HI for message pcssmg is used in or-
der to acelemte the solution Converges e process c
multilevel cpproah coupled with the t additional multi-
g id reproach is taken The resulting algorithm has beer
applied to various ship geometries ad comparisons
with the sequential code solutions Ed experimental re-
suits are presented The results show duct the parallel
version of the f ee surfae UNCLE code sccnrsteh re-
produces these earlier results
INTRODUCTION
Nonlinear, turbulent free surfa flows repre-
sent m import mt cuss of problems wish immediate mr-
val applications e peciclly when mch flows occur in the
vicinity of 9 body These problems are ve y challenging
from c Computational Fluid Dynamical pomt of view m
that Hey dorm Ed c ve y robust m mericcl algorithm,
very large computational resources Ed ve y Urge
limo mts of computing time More import Fitly, in cddi-
tion to robust ess, She CF algorithm must model the
correct physics The original UNCLE ( for UN teddy
Computation of fieLd Equations ) code Taylor (1991),
Whiffield Ed Taylor (1991)) was developed based on
fi st principles to solve She m teddy Rey olds averaged
Navier Stokes (unFtANS) equations without my fm-
ther simplifying less mptions This sequential version
was fmthff extended in Beddhu et cl (1994), (1999)) to
include the elf - to of c f ee surfae ~ Beddhu et cl
(1999) the free surfae go coning equation was fommu-
hted m terms of surfae curvilinear coordi ares
inhoduced on She actual evolving free surfae for the
fi st time Previous efforts have inhoduced the surfae
curvilmear coordinates on c flat surfa The formoh-
tion introduced in Beddhu et cl (1999) allows for the
computation of steep Ed breckmg waves The sequen-
ticl version of She free surfae UNCLE code has been
applied to various geom en ies with c good deg ee of sue-
cess (see Beddhu et al (1999) Ed She references thffe-
in) However, These computations took enommous com-
putmg tame
'.\ nh She motivation of reducmg She r m time
of the sequential UNCLE code when aplied for com-
plex co figurations, c parallel version of She UNCLE
code without She free surfae capability was developed
by Paksjaksh m Ed Briley (1996) it is desigmed to op-
erste at coarse/medi m g am parallelism levels for opti-
m m pe ton once md is based on Single Prog am Mnl-
tiple Dats ~SPhUDI model it .... hlPI for message
passing This version r epic sents one of the fi st parallel
CFD codes that has beer used for solving practical prob-
lems m n rontme marmot N morons test 093-5 w re
used to test She robustness md acuray of She code
Panlrajakshan (I 99 D ~ The effort Pa sented m She ph S-
ent paper incomorstes She fee surfa capability into
the parallel version of She UNCLE code This Evolved
n complete re write of the fi ee surfime code using FOR-
TRAN 90 which is now included 99 n separate module
in She parallel code
The reproach adopted to compute fi ee surfae
flows is to cast She governing equations with respect to
n non memos fi ame so that the fi ee surfime c m always
be made to coincide with n coordinate surfime Thus,
Nervier Stokes equations are 09 t withrespfft to n set of
general unsteady curvilmear coordinates md are solved
OCR for page 806
along with She non Inner free surfa kinematic bo md-
ary condition The modified artfficicl compressibility
method Beddhu et cl (1994) is used to march the solu-
tion m time Approximate inviscid dynamic bo mdary
conditions are used et the free smfae since She wind
stresses ad smfae tension are neglected A two equa-
tion turbulence model is used for closing She m oment m
equations Charateri tic Variable Bo mdary Condi-
tions(CVBC)creusedondhe i flow mdoutflowbo md-
aries
The m mericcl scheme m Tcylor (1991) ad
Whiffield ad Tcylor (1991) uses c finite vol me for-
moktion ad uses the Roe scheme for obtainmg She fi st
order flw.es ad the MUSCL scheme of mL er to ob-
tam higher order flw.es et the cell fan s The flux Jao-
bi ms which are required in m implicit scheme are ob-
tamed m mericclly The viscous flux Jaobias are
obtained using She Shin flyer approximation The result-
ing system of algebraic equations are solved using the
Discretized N wton Rekxation DNR) scheme which
is comprised of Newton iterations with symmetric
Gauss Seidel sub iterations in etch Newton iteration
A Full Approximation Scheme FAS) multig id clgo-
rithm is used to a elemte the convergence The free
surfae kinemtric bo mdary condition is solved using
the same cpproah
Since She present cpproah falls aider the cate-
gory of front tracking methods, it is necessary to regen-
ercte She f ee surfae ad the mderlymg g id et etch
time tep The free smfae is ass med to be of the form
y = Y(t;, t,'t) where to attd Care surfae curvilin-
ear coordinates ad ~ denotes time Thus, in solvmg the
kinematic bo mdary condition one obtains m increment
Ay et every pomt on the free surfa for a increment
At m time Using the Ay's the new free smfae posi-
tion is obtained ad the flow g id is updated using c
btckg o mdg id
The parallel code was verified by mming the
same cases es She sequential code ad the re mlts w re
compared with experiments The majority of She results
show are for Series 60 Cal = 0 6 ad Model 5415
The flow parameters for Series 60 Cs = 0 6 were spe-
cifled by the experiments to be es follows: c Froude
m mber of 0 316 md c Rey olds m mbff of 4 2 X lOs
For Model 5415, the Froude m mber was given es
0 2756 ad She R y olds m mber es 12 02 X lot
Some of She prffUrSO y works m free surfae
flow computations ctout surfae ships w re performed
by Miyata ad Nishimurc (1985), Kodamc (1989), Hmo
(1989) ad Farmer et cl (1993) R - ent works are that
of Taarc ad Stern (1996) ad Beddhu et cl (1994),
(1998), (1999) Taharc ad Stern (1996) use c Poisson
equation for pressure ad use the pressure implicit
split operator PlSCn algorithm to solve She me m flow
equations m d r h B am mdWarmingapproahwi6har-
tfficicl dissipation for solving the f ee smfae equation
They use the so-called finite crurlytic method to discre-
tize She governing equations in addition, they use sepc-
rcte g ids for solvmg She me m flow ad the free smfae
ad use ate polation for trasfening date between the
g ids Beddhu et cl (1994) inhoduced She modified arti-
ficicl compressibility method for solving She me m flow
equations ad use m explicit method for calcoktmg the
free smfae motion For the computation of m m flow
they use c finite vol me, implicit scheme patterning
thei m mericcl scheme after compressible flow solvers
Thus, Roe scheme is used for obtainmg fi st order mm-
mericcl Flare s ad m Leff's MU S CL tppro tch is used
to obtain thi d order corrections The viscous temms are
approximated using second order central differences
ad the time derivative is approximated using ei6'rer
first order or second order backward differences
In contrast to Beddhu et cl (1994), Beddhu et
cl (1999) use m implicit method for calculating the f ee
surfae The method was chosen to be the same es the
one used for She mea flow in addition, c novel way of
tracking She free surfae m time was inhoduced which
preserves She shape of the f ee smfae et etch time step
during various g id operations Since She geometries of
actual ships are ve y complex mch m aprotch is neces-
sary for ndvarming the free surface along curvilmear
coordinates in essence, n bsckg o Ed g id was used
which is fi ed m time G id points on She fi ee surfae are
allowed to move along n particular fam ily of coordinate
Imes, chosen n priori, of She bsckg o Ed g id The por-
tion of the coordi me lines below She fh e surface is then
used to rebuild the actual g id it the next time level os-
ing6he arciengh di traction of the scrusl g id Ime at the
current time level
GOVERNING EQUATIONS
The governing equations in 6 is work are cast
with re pect to n non menial frame in tensor mvari at
form the contmoih equation m She modified stffflcitl
comphssrbility method Beddho et al (1994), (1999))
is given by
P + d div o = 0 (1)
When 1. is She a~tfficinl compressibility p lrtmeter
The moment m equation for viscous, mcomphssrble
flows in n non memos fi tme of reference, in n g avita-
tiomrl field, in non dim ff siorLtl, vector invari at form
isgivenby Beddhoetal (1994),(1999))
,h + V · [' it + P I - R '1] = 0 (2)
OCR for page 807
In Eqs. (1) and (2), u = u*/UO, is the non-dimension-
al absolute velocity vector, v = u + w is the non-di-
mensional relative velocity vector, w is the non-di-
z ,~
direc-
tion ~ = constant surface
~ increasing from the
inflow boundary to the
outflow boundary ~
Fig. 1. Schematic illustrating the Cartesian and
curvilinear coordinates chosen.
denotes that it is a vector. The Stokes tensor is given
by
~ = ~ ( V u + V u ) (3)
where, ~ = p*/~0, is the non-dimensional coefficient
of viscosity. The superscript 'T' in Eq. (3) denotes the
transpose operation.
Casting the governing equations (1) and (2)
with respect to a curvilinear coordinate system
( 4, A, 5, ~ ) and using the so-called partial transforma-
tion in which the vectors and tensors that appear within
the divergence operator are expressed with respect to
the underlying Cartesian coordinates whereas the diver-
gence operator itself is expressed in terms of the curvi-
linear coordinates, one can obtain the so-called numeri-
cal vector form which is as follows:
dQ + dF + dG + dH + dFv + dGv + dHv O (4)
where,
lines
~ increasing from the
bottom boundary to
the top ~
S lines
~ increasing from the Fv = ,:
body to the outer
side boundary ~
mensional grid velocity vector, a= tUo/L, is the
non-dimensional time, c, is the Stokes tensor and
P' = P + x/Fr2 where p = (p*—pO)/pOUO, is the
non-dimensional pressure and X is the body force po-
tential due to gravity. ReO is the Reynolds number,
ReO = poUoL / p0, where, pO is a reference density,
UO is a reference velocity, L is a reference length, and,
p0 is a reference coefficient of viscosity. Fr is the
Froude number given by Fr = UO/ ;, where, a is
the acceleration due to gravity. Note that when the pos-
itive y-direction is aligned in the direction opposite to
the gravity vector, one obtains X = y. A tilde over a
quantity denotes that it is a tensor and an underscore
Q=l:
. ,,
u
v
w
F= ,/;
o
()xxgx + ()xy~y + ()xzgz
Oxy~x + oyy~y + oyzgz
Oxzgx + oyz~y + ozzgz
U1 = U4x + V4y + W4z
U(Ul + at) + P ~x
V(Ul + at) + P ~y
W(Ul + at) + P ~Z
u, v, and w, are the components of the absolute veloc-
ity vector with respect to a Cartesian coordinate sys-
tem, oxx, etc., are the Cartesian components of the
Stokes tensor, ~x, gy and ~z, are the Cartesian compo-
nents of the contravariant base vector grad 4. Expres-
sions for G and H are similar to F and can be obtained
from F by replacing ~ by ~ and I; respectively. Similar-
ly Gv and Hv can be obtained from FV. Figure 1 indi-
cates the coordinate systems chosen.
Equation (4) is the system of non-linear equa-
tions to be solved numerically, using a set of physical
and numerical boundary conditions. The physical
boundary conditions include the noslip condition
OCR for page 808
md dynamic free surfae bo mdary conditions
7he kinematic condition used in this work ca
be derived es follows Ass me thm the free surfae is
repr sentedbythefunction˘ = 0 7henthekmematic
conditionimplies Warsi(1993t
'˘ + q. y˘ = 0
In tffmr of c set of lnertlol curvilinear coordim~tes
(˘ = ˘(4, ll, g, t)), Eq (5) c m be w itten as
it + ~ ,~ + ~ >~˘' + u~ ~c˘ = 0
whffe 6he conhavari mt velocity component ~ is de-
fmed below Eq (4) md us md u~ m be similarly de-
fmed On 6he odher hmd, intemms of c set of non lnez
tlol curvilmear coordim~tes
(˘ = ˘(~(t),tl(t),S(tl1); (t = t)l Eq (5) cm be
written as
r + (u + W ) (~_ + (u + w ) t~ +
(u + w ) = 0 (7)
whffe w, ws md w~ are the contravarimt compo-
nents of the g id velocity vector hoosing ~ = y, Eq
(7) becomes
r + (u + W ) (~_ + V ˘ + (~ + w3) ~C˘ = 0
(8)
Up to this point 6he notion of surfae curvilinear coor-
dim~tes is not needed Note 6~t 6here is no reshiction
pla d on the coordim~tes ~, md g in gff rcl,
,(x, y, z, t) md C = C(X, y, z, t) Now, let the sur-
fae ˘ = 0 be r presented by
˘(4,y,g,t) = y - Y(4,g,t) = 0
(5)
(6)
Substituting, Eq (9) in Eq (8), one obtains
(9)
l,~Y + (u + w ) ~ + (u + w ) ~C —v = 0 (10)
In Eqs (9) md (10) the curvilmear coordim~tes, ad g
need to be mtemreted es surfae curvilinear coordi-
m~tes However,, md g are still of the form
,(x,y,z,t) ad g = g(x,y,z,t) 7he velocity
components ~ md u3, ad, 6he g id velocity compo-
nents w md w~ are now surfae contravari mt compo-
nents
Note 6~t Eq (10) is identiccl in form to the one
obtamed by Farmer et cl (1993) How ver,6hey have ex-
1 1 ~
Xo xl X
Fig 2 Br ckingwaverepresentation
Y(x) is multi—vahed for x, <: x <: x,
Y(~) is how ver single—vxlued mtil reenDy
plicitlyass medthat ~, = ,(x,z) mdg = g(x,z) 7he
formohtion inhoduced hffe is completely genercl md
is vxlid es long es Y rffmcms c single vxlued f mction of
~, md g 7hus, 6he fommoktion inhoducedhere is vxlid
for traking br eking waves up to the pomt of reentry
(see Fig 2) 7he conceptual differ nce betwen the
present fommoktion md that of Farmer et cl (1993) is
that Farmer et d (1993) mtrodue the curvilmear coor-
dim~tesoncflctsmfae(ie x phne)whffecsthepr s-
ent formoktion inhoduces them on the atual free sur-
fae
Any book on differ nticl geometry, Warsi (1998) for
example, c m be consulted for obtainmg the metrics of
the surfa curvilmear coordim~tes From 6hese metrics
the smfae contravari mt components of 6he flow ve-
locity md the g id velocity c m ecsily be obtamed es
outlined m Beddhu et d (1999)
Since, Eq (10) is cast m temms of curvilinear coordi-
m~tes, the m mericcl scheme for solving it c m be pat-
ter d cfter 6~t of Eq (4)
Two two~quation turbulence models are
avaibble m the UNCLE code They are the modifed
Shih md L mley k—C model (Shih md L mley
(1993), Ymget cl (1995), Liou md Shh (1996t mdthe
q—c~ model (Coakley, 1983) The governmg equa-
tions of 6he k—C model cre:
( ~r + di | ky ] = di l(F + ~, ~grad k |
+ 2~ 5: Vu - C (11)
C + dj I C ] = div:(~1 + ~ ~grad C ~
+ C Ck 2~5:7u - C.I~C
+ `~ ~'di r[> 5: 5] (12)
OCR for page 809
whffe C, = 1 44, C~ = 1 94 ~, = 1 3
f~ = I —0 22exp~ - ( 6t )~]. R = kv~ in 6his for-
moktion the eddy vir osity is defmed es urual es
~, = C,,f,,p (13)
How v r, f`, md C`, are defned es
f~, = 1,1 —exp[ - ( e~R~ + e~Ri + esRi )l~2 whffe
e~ = 1 7x10~ = 10-9, es = 5x10-~0 md
R~ = 'vYwher yisthedista eoffthewall
Cr. A~ + A:rk/~)U whffe A~ = 4 0,
A: = ~ 6 cos t, d = (1/3) arc cos(, 6 W
S,jSj~S~, S — I ciu, cQ~i
W = . =, S,jS,j, S,j = 2 cixj + c~x, .
md U~ = ,/S,~S,; + n,jS2~ Fi dly, ~ = 1 3 ad
Q - I ~ ciu~ _ cQ~i)
2 ~ cixj c~x,J
The gov ming equations of 6he q - ~ model
g ,~ + `li~l qY ] = `li`~(F + ~ ~grr`] q ]
2~2C~,r~,5 5 - 140>q (14)
g ~i` + di~l o~y ] = div[(~ + ~' ~grnd to ]
+2C C`,5:5-C oJ- (15)
In 6his formulation, q = ~k, o, = t k md the eddy
vir osity is defmed es before as
~, = C,,f,,p (16)
whffe C~, = 0 9 md f`, = I —exp( - 0 0065qy/y)
The odher constats md f mctions ere defned es C~ =
0 92, C, = 0 045 + 0 405f~,, md ~,~= 1 3
In the ebr nee of smfae tension, contimmity of
the shess v ctor aross 6he mte fae is 6he e at dynam-
ic fre smfae bo mdary condition 6~t one needs to im -
pose This condition was origim~lly obtamed by Hi t a d
Shanmon (1968) A efficient way of implementmg
ther exat conditions is outlined in Beddhu et cl (1997)
How v r, in the present st dy e at dynamic bo mdary
conditions are not used Smce the exterm~l tmgentiel
shesses are neglected md only the etm ospheric pr ssure
is considered in 6he nommel shess component md elso
smce the g id near the free surfae is not fme enough to
resolv 6he weak surfae hyer, the dynamic bo mdary
condition hes been epproximately implemented by
m my eubhors for computing ship rehted flows For ex-
emple, commonly used epproximate conditions ere
'lu = 0; C]V = 0; ciw = 0
wher, z is the di ection opposite to the di ection of the
g evity v ctor ~ 6he pr sent work, c charateristic
varietle based epproah is used which c m be outlined
es follows Fi st, ev y temm m Eq (4) is neglected ex-
cept 6he msteedy temm md 6he i viscid term in 6he n
dirfftion The remltmt eq mion is linearized md r~st
in c diagorurl form es outlined in Teylor (1991) Thus,
one obtrins
ciW + A diW = 0
(17)
wher, W = T~' Q is the charateristic varietle v c-
tor, T~' is 6he left eigenvector of the flux Jaobim
,iG/,iQ md A is 6he diagorurl matrix conteining 6he ei-
ger~lues of the flux Jaobim ,iG/,iQ The subscript
'0' in T~ ' denotes thm it is tr eted es c con tmt me-
tri The eiger~lues are giv n by
i~? = T1t + ullK + tly + WIl~
i~ = u11~ + Y IY + wll~ + k~/2 + c ~ (18)
i~~ = u1~ + ylly + wl~~—k/2—c J
whffe,
c = l(ull~ + vlly + wll~ + 11~/2)2 + p(112 + lly + 11~
On c l, = ll,.~ bo mdary which is t pic dly chosen es
the f ee smfae, i~ md i~ ro, i~ is positiv md
i.` is negativ Thus the charateristic varietle W~
needs to be pr scr~bed ad W~ needs to be exhapo-
hted f om withm the computatiom~l domem How v r,
since i~, md i~ are ffO mie cm either extmpohte or
specffy W. md W~ ~ this work, W~ ad W~ are ex-
traohted from wi6hin the computatiom~l domein, md
insteed of prescr~bing W. md W~ one uses 6he condi-
tions 111 + u1~ + Y IY + wl,~ = 0 md p' = y Fr~ A
3x3 mehix is solv d for the v locity components et
eah g id point using W~, W~ md 6he kinematic con-
dition Ev n though, this is m mvir id epproximation
to the exat viscous dynamic bo mdary condition, it
works quite well for 6he cases consider d md v rY
good eg cement hes been obtamed wi6h measured
wave profiles
The m mericel bo mdary conditions are im-
posed on artificiel (outer) bo mdaries which are
OCR for page 810
imtr oduced to truncate the compnhtioncl dom cm to c ti -
nite size so that the resulting problem c m be solved us-
ing c computer The ass mption is thm the artfficicl
bo mdaries are far removed from the physical body that
they won't affect She acuray of the solution in the vi-
cinity of She body in this work She fluid flow is ass med
to be along She positive x direction with She body fixed
et She origm Thus, far upstream of the body one uses c
characteristic variable based i flow bo mdary condition
Ed c characteristic variable based outflow bo mdary
condition far doss nmecm of She body The upstream Ed
down cream bo mdaries are located et lee t et 5 body
lend hs e ah f om the or i am At the far away side bo md-
ary Ed the bottom bo mdary ei6'rer charatffi tic vari-
ctle based i flow or outflow bo mdary condition is used
depending upon the local velocity vector The side
bo mdary Ed She bottom bo mdaries are located et 5
body lengths f om the origm At the z = 0 bo mdary,
flow mmetry is imposed in addition, She compute-
tiom~l domain is subdivided into crbitTcry sub domains
Ed et the bo mdaries of These domains c two point sym -
meby condition is used
The following procedure is used to cdvarme the
solution from time tep n to time tep n+l:
1) Solve for She mterior of She flow tield using the Dis-
creti:D:d Newton Relaxation method
2) Update the bo mdary conditions on She flow vari-
ctles on all surfaces
3) Use the kinematic condition to find She new position
of She free surfae
4) Find the intersection of the bakg o Ed n lines with
the updated free surfae using c neare t point cpproxi-
mation
5) R create She vol me g id from She free surfae ob-
tamed m step 4
6) Obtain the new metric ccetllcienrs Ed g id speeds
7) Update the free surfae dynamic bo mdary condi-
tion once sham This is done since She shape of She free
surfae has ch aged (m step 2 She f ee surfae et the
previous time level n was used) Ed is fo Ed to en-
h mce the stability of She scheme
8) go to step I
An integ cl q mtity of immense practical in-
terest is She total resistance of c ship The total resistance
is cclcuhted using the followmg expression:
F = J p Frgnx g ~gd-dhl
+ I T, + 1, ny + To g 57g do
T. 14 Try = Re (by + vx);
x~ Re ( ~ x)
In the actual computation, She mteg cl is re-
plaed by s mmation over all the surfae g id cells no
is the component of She mit normal m the x direction et
my point on She body The q mtity 7g d d is the sur-
fae elemental are where Vg = ~ g + gy + go Note
that the body is represented by c C = con t mt surfae
everywhere except She tm msom surfae which is c
a, = constant surfae This fat is properly taco mted
for m the results presented
NUMERICAL PROCEDURE
The m mericcl scheme used in 6 is st dy is
similar to thm proposed by Pa Ed Chakravarthy
(1989) Ed is discussed m detail by Taylor (1991), Ed,
Whiffield Ed Taylor (1991) An extensive discussion of
the methodology has been presented by Whiffield Ed
Taylor (1994) applicable to two dimensional flows The
cpproah taken in 6 is work is to solve Eq (4) implicitly
us ing She D iscret ize d Newt on R boat ion ( DNR )
scheme (O tech Ed Rheinboldt (1970), Whiffield Ed
Taylor (1991)), where She flw.es et the cell faces are ob-
tamed using She Roe scheme (1981) with higher order
acuray achieved using the AdUSCL aproah ( m
Leer (1979); Whitheld ad Taylor (1991)) Writing Eq
(4) m discrete fomm,
Q '- Q' +
~1
F +,l—F' + Go —G~ + H -'—H -
+ Fell+,'—F ~ + Gal+' —GO
+ HVi+'—HVi+' = 0
(19)
where Fn+' = F(Qi i'. Q '. Qi+i . Qi+~) Ed so
on Note thm for ~ higher order flux representation
Fn+~i depends on Qi ~i Ed Qi'+~ as w 11 ffEq (19) is
exp aided for each g id cell, ~ system of aIgebmic
equations are obtained in terms of q' t ah g id
cell where qua+' = Q~+'/`g Shictly peaking
F~+'isafunctionofbodh A+' mdthemehicsatn+l
Since the metrics at n+l are k own, no linearization
needs to be done with respect to the metrics Hence
OCR for page 811
Eq. (19) is regarded as a function of qn+i alone. In
functional form, Eq. (19) can be represented as
X~qn + ~ ~ = 0 (20)
Solving Eq. (20) involves finding the roots of
Domain iteration in parallel across processo
~;~;1
3
l 8
tr~c,Restr~
Jacobian, Resin
SGS iteration|
,13
OCR for page 812
achieved on up to 500 processors using appropriately
sized grids of up to 50 million points.
Sub-domain 1 Sub-domain 2
I Forward sweep
Partial boundary
update by
message passing
Lit Lit
Backward sweep
Partial boundary
update by
message passing 4>J L_~)J
Fig. 4. Sequence of Operations for One BJ-SGS/1
Sweep in Asynchronous Mode
MULTIGRID STRATEGY
The multigrid approach (Sheng et al (1995))
used to accelerate the iterative convergence at each mul-
tilevel level is described schematically in Figure 5. It is
seen that a time-dependent solution is obtained through
the use of Newton sub-iterations, which are the most
costly part of the overall work. However, this effort can
be reduced with the aid of an efficient multigrid method
(FAS). The unsteady multigrid method is basically the
same as the steady, except that both the fine grid and
coarse grid equations must be solved at the same time
level to ensure temporal consistency, while in the steady
multigrid approach, time is advanced in the fine grid as
well as the coarse grid to achieve full efficiency. The
two-level multigrid method for a general unsteady
equation can be briefly described as follows:
by Newton's method.
1. Iterate Nh(Qh)=o J\E times on the fine grid h
2. Restrict the residual and solution to the
coarser grid 2h, and iterate N2h(Q2h)=~2h ~ times
where r24N2h(I2hh Qh)—R2hh (NhQh) is the relative
truncation error between the grids h and 2h.
3. Interpolate the correction from the coarser
grid to the fine grid and update the solution
Qh=Qh+Ph2h (Q2h—I2hh Qh)
_=
multigrid cycle |
| Iterate Nh(Qh)=0 on fine grid h | Oh
Restrict the residual and solution
to the coarse grid 2h
Compute R=Nh(Qh)
Compute I2hh(Qh)
r24 N2h(I2hh Qh)—R2hh (R) |
| Iterate N2h(Q2h)=~2h on coarse grid 2h |
I Q2h
~ c2h
I Interpolate the correction to fine grid l
I Compute Ih2h (A o2h) l
| Update fine grid solution |
| New oh l
Fig. 5. Schematic of the Multigrid Cycle
4. Repeat steps 1~3 for ~ times at the same
time level, using Oh as the new approximation to Qn+~.
In the above procedure, J\E is the number of
Newton sub-iterations for the fine grid and coarser
grids, and ~ is the number of multigrid cycles imple-
mented at each time step. Choosing different values of
J\E and ~ may form different multigrid strategies and
result in different effects. However, an important fact
is that the cost of CPU time is proportional to the multi-
grid cycles (my) at each time step.
GRID GENERATION
The grids presented in this work are generated
using Graphical Unstructured Multi-Block (GUMB)
structured grid system (Jiang (2000), Remotigue
(1999), Jiang and Remotigue (1998)) which is being de-
veloped in house. GUMB retained and further enhanced
both the initial General Topology Model (GTM) data
structure and geometry engine based on Non-Uniform
Rational B-Splines (NURBS) which were originally de-
veloped for NGP (Thompson (1992), Remotigue
(1994)). They are both coupled with a structured grid
OCR for page 813
generation library. A schematic of the organization of
the libraries of GUMB is shown in Figure 6.
Fig 6. GUM-B Schematic of Library Layout
GUMBO, Graphical Unstructured Multi-
Block Omnitool, is a graphical user interface, devel-
oped at Mississippi State University, that has flexible
and general repartition and manipulation algorithms
that automatically account for the boundary conditions
and connectivity. Furthermore, the application of
boundary conditions and connectivity is simple and
straight forward. In addition, grid assurance and quality
measures are included to validate the grid, boundary
conditions, and connectivity.The schematic of GUMBO
can be seen in Figure 7.
Transformation Tools
Repartition Tools
Indice Operations
Quality Assessment
Fig. 7. GUMBO Schematic of Library Layout RESULTS
For all the cases presented, load balancing
through domain decomposition, specification of bound-
ary conditions and automatic detection of inter-block
boundaries were done using GUMBO.
For nonlinear free surface calculations it be-
comes necessary in the present method that the flow grid
be adapted to the free surface such that the top surface
of the flow grid coincides with the free surface. Beddhu
et al (1998b) took an intersection approach where the
every reline of the background grid is intersected with
the actual free surface to redistribute points on the free
surface in order to generate the flow grid for the next
time step. Thus, this approach preserves the shape of the
free surface during the grid regeneration process and is
suitable for unsteady free surface flows. Since the pres-
ent work is focussed on steady free surface flows, the
following approach is taken to redistribute points on the
free surface: For each point (i, k) on the free surface find
the closest point on the corresponding reline of the
background grid. Note that this approach distorts the
free surface. Since only steady state is of interest, one
expects the free surface to move little as the iteration
count increases and thus one expects the free surface to
reach the same shape as it would have reached via the
intersection approach. However, in practice, though
very little free surface movement is noticed at large it-
eration counts, the final shape is slightly inferior to the
one obtained through the intersection approach. Further
testing and an intersection approach based on NURBS
are underway.
0.010
0.000
-0.010
-0.020
-0.6 -0.4 -0.2 0.0 0.2
x
0.4
Fig. 8 Hull Profile Comparisons for Wigley Hull
Fr= 0.289; Re= 3300000
Expt Sequential - Parallel
The free surface version of the parallel UN-
CLE code described in the earlier sections was used to
compute the free surface flow fields around three popu-
lar hulls: (1) Wigley, (2) Series 60 CB = 0.6 and (3)
Model 5415. These results are presented below.
Wigley hull is a fairly simple geometry and has
an analytical description to it. Referring to the coordi-
nate system in Fig. 1, the equation for the hull is given
OCR for page 814
Y 2 L 1 - — , where Bis
{ Or){ to)
0.01
y
O. _
-0.01
-0.02
\ ram ~
. . . . . . .
-0.5 0 0.5 1 1.5
x
Fig. 9 Comparison of Hull Wave Profiles for
Series 60 CB = 0.6
Fr = 0.316; Re = 4.02 x 106
Expt Sequential -- Parallel
0.0135
y 0.0085
0.0035
-0.0015
0.0135
`7 0.0085
0.0035
-0.0015
0.0135
0.0085
0.0035
-0.0015
., .. ....
x = 0.000 `= 0.050
to 3 t~ 3 the
x.-0.1.00, x.-0.1.25, x.-0.1.50,
....
x=0.~5
. ~
.... _
0 0.05 0.1
z
Fig. 10 Transverse Wavecut Comparison in the
Bow Region for Series 60 CB = 0.6
Fr = 0.316; Re = 4.02 x 106
· Expt Sequential -- Parallel
....
.x = 0.200
.
. . . .
0 0.05 0.1 0 0.05 0.1
z z
....
.x = 0.250
2.> ,
the breadth, L is the length and D is the draft of the ship.
The following ratios were taken L/B = 10 and
L/D = 16. The flow conditions are Fr= 0.289 and
Re = 3.3 x 1 o6. Computed results using the sequen-
tial UNCLE code for the Wigley hull were reported in
Beddhu et al (1998a). Hull profile for Wigley hull is
shown in Fig. 8. Note that the parallel code produces a
1.50e-02 . . . .
. x = 0.850
5.00e-03 :
-5.00e-03 . . -, ., n
1.50e-02
: x - 0.975
y 5.00e-03
-5.00e-03
0.0135, Mix- l.030
y 0.0085 : `',
0.0035 .
-0.0015 . . . .
0.0135 x- 1.200
y 0.0085 ; .
0.0035 ~ '
-0.0015
0 0.05 0.1
- x-l.10'0-
~ i,
....
0 0.05 0.1 0
. x=0.950;
,
· x-1.1504
A' ~
. . . I
0.05 0.1
Fig. 11 Transverse Wavecut Comparison in the
Stern Region for Series 60 CB = 0.6
Fr = 0.316; Re = 4.02 x 106
Expt Sequential - Parallel
-C .5
- ~ , , I ~ , , E,xpt ~ , , , I ~ , , I , ~
~ O.5 1 1.5 ~
X
Fig. 12 Comparison of Experimental and
Computed (Parallel UNCLE) Wave Contours
forSeries60CB=0.6
Fr = 0.316; Re = 4.02 x 106
profile that is in very good agreement with the sequen-
tial code and the experiment.
Series 60 CB = 0.6 is a geometry for which
a lot of experimental data is readily available (Longo et
al (199311. The flow conditions chosen for running the
parallel UNCLE code were as follows: Fr = 0.316 and
Re = 4.02 x 106. Results using the sequential version
OCR for page 815
of the UNCLE code were reported in Nichols (1998) and
Beddhu et al (1998c). Comparison of the hull profile be-
tween the sequential and parallel versions with the ex-
periment is shown in Fig 9. It can be seen that the predic-
tion of the parallel version is similar to the sequential
version. This similarity can be further seen in the trans-
verse wavecuts presented in Figs 10 and 11. In Fig. 12,
comparison of the overall wave contours between ex-
periment and computation using the parallel UNCLE
code is shown. This figure shows that there is reasonable
overall agreement. In Fig.13 the u-velocity con-
~ ~ ~ I I I I I I I I I I B.:.:.:.:.:.:.:.:.~:.:.:.:.:.:.:.:.:i.:.:.:.:.:.:.:.:.:.~:.:':':':':':':':i':':':':':':':':':E':':':':':':':':I':':':':':':':':':t':':§ ~1~1~1
· -~.O 5 -~.07'5 ~ O.07'5 0.O 5
Fig. 13 Comparison of Experimental and Com-
puted (Parallel UNCLE) u-velocity contours on the
Propeller Plane for Series 60 CB = 0.6
0.020 1
~ 010
z
-0.0 1 0
-- O Expt (DTMB)
3' . 1 1 1 1
-0.5 -0.3 -0. 1 0.1
X
1 1 1 1 1 ~
0.3 0.5
Fig. 14 Comparison of Hull Wave Profiles for
Model 5415
Fr = 0.2756; Re = 12.02 x 106
tours on the propeller plane of the parallel code is
compared with the experiment. Here, it can be seen that
O.~05
O.~04
O.~03
o.~2
O.~01
-. 1
-~.~02
-~.~03
-~.~04
-I. OO 5
jibe
O.5 1
X
Fig. 15 Comparison of Longitudinal Wavecuts
at z = 0.0965 for Model 5415
Fr = 0.2756; Re = 12.02 x 106
Expt Sequential - Parallel
Fr= 0.316; Re = 4.02 x 106
O.E
0.4
-0 .7
-G 4
Fig. 16 Comparison of Wave Contours for
Model 5415
Fr = 0.2756; Re = 12.02 x 106
the agreement is quite good. A similar agreement for the
sequential code is shown in Beddhu et al (1998C).
The next test case to be presented is the Model
5415. Extensive experimental results for this model is
reported in Ratcliffe and Lindenmuth (1990) and Oliv-
OCR for page 816
eri et al. The flow conditions are Fr = 0.2756 and
~.02
-~.04
O.~S
Computation
Fig. 17 Comparison of Contours of U-Velocity
Component at the Propeller Plane for Model 5415
Fr = 0.2756; Re = 12.02 x 106
0.014
0.012
0.010
0.006
CT
x
0.004
0.002
0.000
~ Experiment
Sequential UNCLE
Parallel UNCLE
'IVY
0.0 1000.0
2000.0 3000.0 400C
ncyc
Fig. 18 Resistance Comparisons for Model 5415 CONCLUSION
Fr = 0.2756; Re = 12.02 x 106
Re = 12.02 x 106. The computed wave profile has
been plotted against the available experimental wave
profiles in Fig. 14. Note that the experimental wave pro-
files themselves do not fall in each other's uncertainty
limits. Thus, the implication of such disagreements be-
tween experimental results to the efforts for verification
and validation of numerical results is not clear. Howev-
er, the computed profiles agrees reasonably well with ei-
ther measurement. In Fig.15, the comparison of the lon-
gitudinal wavecut at z = 0.0965 between experiment and
the two versions of the UNCLE code is shown. It can be
seen that the agreement is quite reasonable. The overall
comparison of the computed and experimental wave
contours shown in Fig. 16 also shows good agreement.
Fig 17 compares the contours of u-component of veloc-
ity between the parallel version and the experiment. It
can be seen that the agreement is excellent. Computed
and measured total resistance comparisons are shown in
Fig.18. It can be seen that though the sequential and par-
allel versions follow different paths of convergence they
both converge to the experimental value in about 4000
cycles. The difference in the paths of convergence is due
to algorithmic differences between the two versions. For
example the sequential version has no multilevel capa-
bility. Finally, the computed and measured stern wave
patterns are compared in Fig. 19. Since the transom is
wet at this Froude number, computing and matching the
stern wave system is supposed to be a tough challenge.
It can be seen that the parallel version of the UNCLE
code does a very good job of predicting this tough flow
feature. A similar comparison with the sequential code
is shown in Beddhu et al (1998b, 1999~.
Fig. 19 Comparison of Stern Wave Contours for
Model 5415
Fr = 0.2756; Re = 12.02 x 106
· Expt Sequential - Parallel
The free surface version of the parallel UN-
CLE code was designed to be a production mode code
that is fast, robust and reliable. It has been tested against
various geometries and has produced similar agree-
ments with experiments as the free surface version of the
sequential UNCLE code. A NURBS based intersection
algorithm is being developed that will improve the ac-
curacy of the computation of free surfaces. This code
has been successfully transitioned to DTMB and has un-
dergone extensive user trials. Currently, efforts are be-
ing focussed in solving unsteady free surface flows.
ACKNOWLEDGEMENTS
This work was supported by grant
N00014-97-1-0959 from the Office of Naval Research.
The grant monitor is Dr. Edwin Rood. This support is
greatly appreciated.
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OCR for page 817
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OCR for page 819
DISCUSSION
S. Cordier
Bassin d'Essais des Carenes, France
he use of gods fitted to She free surface has
limes when s~muhtmg non-linear tree su face
flows (braying Ed jets for example) C m you
tell us what your pl ms are to add ess this
AUTHOR'S REPLY
Representative terms from entire chapter:
free surfae
