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OCR for page 897
An Unstructured Multielement Solution Algorithm
for Complex Geometry Hydrodynamic Simulations
D. Hya ns, K. Sreenivas, C. Sheng, S. Nichols, L. Taylor, W. Briley,
D. Marcum, D. Whitfield (Mississippi State University, USA)
ARSTRACT
The primary objective of this st dy is to demon once
m efficient incompressible -I w sol ff capable of per-
fommmg time~ccumte, viscous, high R y olds m mbff
flow simulations for complex geometries using general
mshuctmed g ids This pcmilel flow solver is demon-
shated for large-sccle meshes with viscous mbicyer res-
olution if/ ~ 1) up to /) = IU° Ed approximately
10' pomts or more The prolate spheroid is presented
es c model problem with complex -I w ph nom ergo sur-
fcce pressure di tr~butions cg ee w 11 with experimental
data Realistic applications Include 1) the NOAA RV-
40 ship hull, 2) he SUBOFF hull et model scale Ed f 11
scale conditions, Ed 3) the DTMB 5415 hull m norup-
pended mddynamicf Ih cpp nd d co flgmations in
cil cases, cg cement between computations Ed experi-
mentcl measurements r mges from reasonable to excel-
lent
INTRODUCTION
The primary objective of 6 is st dy is to develop
Ed demon once m efficient RANS mcompressbble -I w
solver capable of perfommmg time-accurcte, viscous,
high Rey olds n mber flow simulations for complex ge-
ometries using mshuctured g ids This flow solver is
to be dem on trcted for large-sccle multielement meshes
with good subicyer resolution :~y ~ 1) up to }'. = 10'
Ed approximately 10'i pomts or m ore with m emphasis
toward hydkodynamic applications Sample results are
sh wn here for bodh surface ship Ed submarine geome-
tries
The present solution cigori6 m is related to se It
previous effo ts; implicit algorithms for flows on m-
shuctmed g ids have been inve tigated extensively by
Variety of mthors [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
The current approach is m evolution of the implicit -I w
solver Ed code of A der on et cl [11] [12] [13]; She
solver developed m this series of works demon omies
3D, implicit, high Rey olds m mbff solution ccpabil-
ity Also, this work follows She mstructured mult~block
solver of Sheng Ed Whitfleld [14] [15] which uses She
same core solver but employs c mult~block technique to
reducememoryconsumptionby70% These tudiescre
in turn related to he multiblock structured solvers origi-
m~tmg f om Taylor, '.\~etk id, Ed Sheng [16] [17] [18]
Elements of he present approach to parallel solution are
rented to he parallel multiblock shuctmed g id solver
of ParJccjaksh m Ed Briley [19] The present parallel
mstructured viscous -I w solver is based on c coarse-
g cmed domain decomposition for concurrent solution
withm subdomcins cssig cd to multiple processors
All tetmhedLcl Ed multielement mshuctured
meshes in this work are generated with m cdvarm-
ing normal methodology for he bo mdary layer ele-
ments, md m AFLR cdvarmmg f ontflocal rffom c-
tion methodology for he isohopic elements es given
in [20] This procedure allows for the generation of
high quality mshuctmed g ids suitable for simulation
of high R y olds m mbff viscous flows All geometry
preparation md surface g id generation is performed us-
ing SohdLhl sh [21] with AFLR surface g id ge fiction
[22]
This pCpff is orgmized es foil ws: the govem-
ing equations are outlined, follow d by the m mericcl
procedures for he discrete solution of These governing
equations Solutions of high Rey olds n mber flows
aro md6 eecomplexhydkodynamicco flgmations Ed
c smaller model problem are presented next to demon-
stmte She efficiency md accuracy of the solution tech-
nique Concktsions are s mmarized in the host section
GOVERNING EQUATIONS
The unsteady th ee-dimensiomtl mcompressbble
Rey olds~raged Navier-Stokes equations are pre-
sented here m Cartesi m coordinates md in conservative
form Foil wing [23], m artiflcicl time derivative term
(rDp<~/r91, where p = Of) ht. been added to the con-
tmnity equation to cast She complete set of governing
e quet ions mt o e time -marchmg form The nondim en-
OCR for page 898
<9~`Z``v—| F ZtZ.Z=—~ G Zia.3 (1)
'91 IZ ..ZZ ~1' ~'SZ
where Z ~ is 6he o rtWii d pointing mit normli I to 6he con-
trol vol me V he itor of dependent variiibles imd
the compo' mts of the iZ iscid imd visco n flux Zctorr
s eg~ m i s
LP1
LZZ'.]
- il ((9 - (t, ~
F r ~ = u(9—Z9Z r P
.'(9-IZ P
_ "'(9—f ~P _
G ,¢= ~ tZt~ Zrl _^Y It— l (4)
r¢:r~r—Ztyr,y—Zl,r,.
where il is 6he s tiflciii I compressibility ps Zmeter (typ-
ically 15 in this work), u, ~, imd ~r s e 6he Cartesi m
docity compo' mts in the r, o, imd, directions, imd
ft.:, fLy, imd f`, s e 6he compo' mts of the normi liZvZd
conhol d meflice ZCtor (9is6he docitynormlilto
;; conholvol mefli >:
(9 = f, tt - fty~'- f ,~'- `~' (5)
where the g id speed a' —(l',rl:—l ;~fLy—1 Z)~)
Note thtit l ~ = l 'ri—lyj—l' k is 6he control vol me
flice docity hevariiiblesindhepre Zdingequationr
s e nommli liZvZd with resp it to ii chti ii iteristic length
scii le (L) imd fi yerheirm values of docity (` ~ ~), den-
sity p:,), imd vi iosity q' ~) h n, the R y olds m m -
ber is deflmed ii si 1~ = i -, L/I., PZ ZSS~Z is nommli 1-
iZvZd wi6h P = (P~—P,)/p ,i -,c, where P* is the
local dimensio'~I sh tic pZ Zssure he visco n tresses
gi m in Equation 4 s e
ri, = (1t—1~) ( 9;-; OZ i)
(6)
where /mmd /q iimZ 6he molec br imd eddy vi iosities,
re pecti dy
NUMERICAL APPROACE
he bli seliZ Z flow sol x is ii node-centered, flmite
vol me, implicit sch me ii pplied to gff xti I mstZ itured
g idswi6hnonsimpliciilelements heflowvariiiblesti e
stored s t the Z tices imd ~ fli cc mteg s Is iire evaluated
on 6he mediim dual surro mdingeiich of 6hese Z ti yr
he nono xlirpping control vol mes fommed by the me-
diim dum completely co x 6he domli in, imd form s mesh
thtit is dual to 6he elementtil g id h n, ii o' Z-to-oZ Z
mlippmg exists betweff` the edges of the origi'~I g id
imd the faes of 6he conhol vol mes
he sol Ztion ii Igorithm consists of 6he foil wmg bti -
sic steps: reconsh ition of 6he sol rtion tti tes ii t the con-
trol vol me fli ies, evaluation of the flux integ ii Is for
eiich conhol d me, imd the e d rtion of 6he sol rtion
(2) in eii ch control d me m time
A higher order spti tiii I medhod is consh ited by ex-
(3) trtipolliting 6he sol rtion st the ffticer to the fli ZS Of
the surro mding control vol me he unweighted least
squares method (sol Zd vis QR flictori:~tion [12]) is
ned to comp rte 6he g s dients iZt 6he Z tices for 6he ex
trti polli tion Wi6h these g s dients k ow `, the varia les
lit 6he mte fa Z s e comp rted with s flkst ordff Tiiylor
series s s
Vr = Qo - ~Vo t (2)
whe' Z t'is ii ZCtor 6~t extends from node 0 to the mid-
point of the edge s ssociii ted wi6h 6he control vol me fa Z
in question
Re Zi dual Evah ad m Z
The go xnmg equations s e di xetiZvZd rsmg s fl-
nite ol me tffhmique; thn, the surflice integlils m
Equation I iimy iipproximlitedby s quadkiiture o x the
surfli ceofthecontrol ol meofmte'yst So,themmff-
iciil discreti: rtion of the splitiiEl temms iissociiited wi6h
the conhol ol me suno mding fft x 0 Z ZS Zlts m
<9~1U _ ~Z = 0
(8)
whe' Z the spiZtii Z residual ·l contti ms s 11 contrib rtionr
from the discrete s pproximli tion to the mviscid imd vis-
co n terms C~ = ·l ;~ l ,.j~) Also, 6he q mtity q is
deflmed ii s `1 = 16 Z (?`Zv
Sptitiiil R sidual
The evaluation of the discrete residual is pfffommed
sepli iitely for 6he mviscid imd visco n terms gi m m
Equation I The Roe Zheme [24] is ned to evaluate
the mviscid flwxZs lit eah fa Z of the conhol vol me
The s Igebrti ic flux ZCtor is repkced by s m mericii I flux
fumytion, which depends on the recon tr Zted diitti on
eiichsideoftheconhol ol meflice:
4=.>X(~?I.)-F(~?~)~-.~-i(~?ll-~?l.) (9)
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Representative terms from entire chapter:
nominal wake
where .i = f .if I The mlitri 1t is ii mlitrix con-
sh cted from the right eiger~ctors of 6he flux Jao-
bia, imd ~ is s diiigmurl mli trix whose enhies contti m
the iibsolute values of the eiger~lues of the flux Jao-
bia The eigensystem used in this work is bli sed on
thm reported in [18] Note 6~t i is evaluated wi6h Roe-
ii rti ged variiibles, which is simply the iFu i6 metic s r-
s ge betw en left imd right solution states in the cs se of
i compressiblefl ws
Viscous flw.es on tehtihedLal meshes s e eval med
usmg s flmite- ol me s pproli ch, which is equivalent to s
Gil lerkin flmite element method in 6his techmique, veloc-
ity g iidients s e evaluated in eiich tetrtihedkiil element
imd the viscosity is computed s s 6he s rtige of the nodes
mcking up the element With this i formli tion, 6he vis-
cous flux ctor is eval med imd 6he re mlts cii ttered to
the nodes comprising the element
For genera I element g ids, it is expedient to use only
edge-locii I i fommlition to compute 6he viscous flwD:r
This iillows 6he evaluation of viscous flw.es on eah
fli ce of the control vol me without regliU d to 6he vary-
ing element t pes of the mesh An s Igori6 m in which
no element i formlition is used outside of metric com-
puttitions is tffmed s "g id himsplim nt" s Igorithm [25]
To 6his end, 6he viscous flw.es s e evaluated di ectly lit
eiich edge midpoint usmg sepli iite aproximlitions for
the normli I imd timgentiii I components of the g s dient
vector to consh et the velocity dffivii tives [26] Usmg s
directiom~l derivii tive s long the edge to s pproximli te the
normli I component of the g s dient imd 6he s rtige of
the nodii I g adients to s pproximli te the t mgentiii I com-
ponent of 6he g s dient leii ds to the foil wing expression
[27
?i~ ~ ~Q [Q. - t?i - i 3X] 13 l3 (10)
The w ighted leii t squares medhod is used to evaluate
the nodii I g s dients in 6he prff eding fommulli
Temporti I R sidual
After the splitiiil terms htive been suitably dis-
cretived, the time derivii tive temm s ppeii ing in Equation
8 must be s pproximli ted A generti I di fere ce expres-
sion is ii~illible for 6his purpose [28] [29], imd is given
s follows:
,, 9~ \1 t9¢~q"~_ ~1 of q") 63 \ ,,
q 1-6, <~{ :-t, of 1-6, q
- - - (11)
where 41'' = q" ~—q" A flkst order acurti te in time
Enler implicit scheme is given by the choices 9~ = i,
63 = 0 Correspondmgly, s second order time acurti te
Enler implicit scheme is given by 6 ~ = i, t93 = 1/2
Si ce 6 ~ = ~ for both time discretizations used in 6his
work, Equation I I cim be fm6her simplifled:
~—930f (q ) ~_6341" (12)
Using Equation 8 to repkce the time derivii tive,
~q -, ', ~q =- 16 'R" ~ (13)
By 6he deflnition of q, one cim w ite q = JV,
where Jisthe ol meii rtigedsolutionvariiiblevector
1/V 1 ' , (?' V Then, 6he following two identities cim be
formed:
~q'7=v'7 '~?~-~?~V'7 (14)
~q'7 '=v'7 '~?" '- ?''~V'' ' (15)
Inse tmg the iibove two identities mto Equation 13, one
s rives s t the following expression:
v'7 '~?" - ,~,,V'' '~?'1 '_
'?'1 [ ~1' ] _ ~ _ 9 '~" ~ = U (16)
Now, one must consider 6he Geomeh ic Conservii tion
Lii w (GCL) This tti tement rektes the rs te of ch mge of
s physical vol me to the motion of the vol me faes:
V=./~ ]~`V=./ ]~ 74rtt (17)
0t S] ,')
According to Thomli s imd Lombli d [30] imd is ter Jii ms
[31], the solution of 6he vol me conserviition equation
must be perfommed in exatly the ss me mii mff s s the
flow equations to ensure 6~t GCL is ss tisfled This pro-
cedureensuresthmspurioussourcetemms msedby ol-
me ch mges iim elim incted Using 6he ss me time di fer-
e cing expression Equation 12) to iipproximlite Equa-
tion 17,
~V"- ,~(, ~V" ~ ~
—63 <~ (18)
~:i ~i=~'U]leui ~
OCR for page 900
Time Evolution
A Newton itffii tive time evolution scheme is ii pplied,
which requi es the solution of ii pli se Imeii system ii t
eii ch nonl inear sub iterti ti on:
I ''' <9~°" 1 ~?'' I ''' (20)
where
t:_63'vo' 1~?o-6 vO ·?o
-?'''~u'
~ [CWlfw—Cblft2] [V] +
R {V [(v + (1 + Cb2 ) v) Vv] —Cb2 TV · Vv}
(28)
lit = PVfvl (29)
The constant and secondary function definitions are
given in t321. For more accurate vortex preservation in
the field, a modification to the production term is avail-
able as described in t151. The discrete version of the
transport equation is solved in an Euler implicit fashion
as described for the mean How.
The two equation q—w turbulence model is also
available for simulation of turbulent effects in high
Reynolds number Hows. This model uses a transport
equation each for the velocity scale and length scale to
specify the distribution of the eddy viscosity. The ve-
locity scale is defined as the square root of the turbulent
kinetic energy (qT = 4). The length scale fit is then
defined by a relation between qT and w (fit = qT/W)-
The eddy viscosity is given by the standard definition
(shown here in nondimensional form):
fit = Cat D—Re
w
The field equation for the variables qT and w is given in
Equation 31 and Equation 32 in nondimensional form.
Notice that the time rate of change of the turbulent trans-
port variables are made up of standard contributions
from convection and diffusion, plus an additional source
term that models production and destruction of the vari-
able.
0t + V · (qTV) — R V [ (A + p ) VqT] = Hq
(31)
At +V (TV)—ReV [(it Pro) ]
(32)
where the source terms in the field equations are defined
as:
Hq =
2 [COEDS - 3 `~' —1] wq
~ [C1 (Cat w2 - 3 is, - 1)—C2] w2
where )° is the strain rate invariant and ~ is the diver-
gence of velocity (taken to be zero for incompressible
Hows). Constants are taken from the version II q—w
model given in t331.
The diffusive terms in both turbulence models are
discretized in the same manner as the viscous terms
| In~tiali:ze Field ~~ [Q. VQ,
_ - [v, pt]
Time sted ~
[
r
Compute VQ
Compute ~
Compute [A]
I ;~ [E ~ I it ~ ~ t~a
Matrix-Vector Multiply
Update i\Q
bu`~ iFb~1
Figure 1: Iteration hierarchy used for the parallel un-
structured solution algorithm
for the mean How, and the convective terms are com-
puted via pure unwinding. Appropriate consideration
is given to maintain positive operators in the formation
of the Jacobian matrix for the implicit solution of the
transport equation(s). The respective turbulence mod-
els are incorporated with the mean How solution in a
"loosely-coupled" procedure; that is, the core governing
equations are solved first, then the turbulence model is
solved independently. This procedure allows for easy
interchange of the turbulence models.
PARALLELIZATION
For quick turnaround time in a design environment,
it is essential to parallelize the How solution algorithm.
The present parallel unstructured viscous How solver is
based on a coarse-grained domain decomposition for
concurrent solution within subdomains assigned to mul-
tiple processors. The solution algorithm employs itera-
tive solution of the implicit approximation, with the con-
current iteration hierarchy as shown in Figure 1. Also,
domain decomposition takes place with each node in the
domain uniquely mapped to a given task. The code em-
ploys MPI message passing for interprocessor commu-
nication.
In general, the parallelization of an existing vali-
dated How solver should satisfy several constraints. First
and most important, the accuracy of the overall numeri-
cal scheme must not be compromised; i.e., the solution
computed in parallel must have a one-to-one correspon-
dence with the solution computed in serial mode. For
the current numerical algorithm, this ability has been
(A
o
(A
~A6
10~
105
104
103
1
ideal run time
predicted run time
· actual run time
-
\~
\
10
CPU count
Figure 2: Actual versus ideal execution times as a func-
tion of the number of processors utilized; predicted run
times are from a heuristic performance estimate de-
scribed in t341.
shown in t341. Also, the code must be efficient in its
use of computational resources. This characteristic is
measured in terms of memory usage and scalability, as
well as the fact that the parallel code should degenerate
to the serial version if only one processor is available. A
sample scalability result is shown in Figure 2. Finally,
the consequences of the inevitable domain decomposi-
tion should not seriously compromise the convergence
rate of the iterative algorithm.
APPLICATIONS
A key contribution in this work is to demonstrate this
solution methodology on realistic hydrodynamic appli-
cations. To this end, several complex configurations are
shown. Along with the capability to handle complex ge-
ometry high Reynolds number Hows, the ability to rotate
actual propulsors is demonstrated in the following re-
sults. This enabling technology allows a true picture of
propulsor-hull How interactions in an unstructured CFD
context.
Prolate Spheroid
The prolate spheroid is a surface of revolution which
produces a relatively complex Howfield even in steady
How at moderate angles of attack. Regions of laminar,
transition, turbulent, and separated How all are present;
in addition, each are functions of the angle of attack.
Further, the body itself loosely resembles a submarine
hull, which has obvious relevance for a solution algo-
rithm with a hydrodynamic focus. Experimental data is
available from Meter t351 for 3D boundary layers devel-
oping on the prolate spheroid; this data is in the form of
surface pressure distributions for steady Hows at angle
Figure 3: Pressure distribution on body surface and at
C/L = 0.83
of attack and for unsteady pitch/plunge/turn body mo-
tions. Only the steady results are discussed here; un-
steady body motions are to be the subject of future study.
The How condition for the prolate spheroid case pre-
sented here is a Reynolds number of 4.2x106 based on
the body length. The grid utilized consists of 750,000
points, 1.2M tetrahedra, and 1M prisms. Normal grid
spacing is set at 10-6, which leads to y+ values of ap-
proximately 0.4 for the first point from the surface. Re-
sults given in this section are computed with the second
order q—w turbulence model.
Figure 3 shows the pressure distribution on the sur-
face of the prolate spheroid and the pressure distribution
on a cutting plane placed at C/L = 0.83. The stagnation
point is visible underneath the nose of the spheroid. The
most striking feature is the separation line extending the
length of the body where the boundary layer is unable to
remain attached to the surface due to the high angle of
attack. The separated boundary layers roll into a vertical
structure on the leeward side of the body, which causes a
secondary low pressure line where the vortices impinge
on the leeward surface.
A comparison of the computed results and measure-
ments is shown in Figure 4 and Figure 5; note that a
circumferential angle of zero corresponds to the wind-
ward side of the prolate spheroid. Overall, agreement
with experimental data is very good, with the How trends
shown by the measurements being reHected in the com-
putations. The effect of the primary vortices on the body
pressure (0 ~ 155°) is underpredicted somewhat by the
computations at C/L = 0.44 and again at C/L = 0.56
(not shown); however, all other stations at which com-
parisons may be made demonstrate excellent correlation
between computed and measured data.
0.6
0.4
0.2
o
c)
-0.2
-0.4
-0.6
-0.8
0.4
0.2
o
c)
-0.2
-0.4
-0.6
-0.8
asp Experimental data
~ —Computations
~ x/L = 0.1 1
°\
o\ /o
0\~/,3OO
°:oo~° °
:
~ o\ /o
~ °~o°
3 °o~°°
0 30 60 90 1 20 1 50
circumferential angle
x/L = 0.31
x/L = 0.23 =
~ ~°
o\ /o
~^,°
If )
..,..
x/L=0.44 >
it\: ~ an':
1 80 30 60 90 1 20 1 50 1 80
circumferential angle
Figure 4: Unstructured algorithm compared to experi-
mental data; stations C/L = 0.11, 0.23, 0.31, 0.44
0.6
0.4
0.2
o
c)
-0.2
-0.4
-0.6
-0.8
0.4
0.2
o
c)
-0.2
-0.4
-0.6
-0.8
: Experimental data
Computations
x/L=.69 =
x/L= 83
~~!
°oo°~
x/L=.83
'a
~ _
:
30 60 90 1 20 1 50 1 80
circumferential angle
:~ x/L=.77
°°
too
GDo:D
I x/L=.9O
4=~=-
/ )
3
30 60 90 1 20 1 50 1 80
circumferential angle
Figure 5: Unstructured algorithm compared to experi-
mental data; stations C/L = 0.69, 0.77, 0.83, 0.90
Figure 6: Surface grid for the fishing vessel case; exper-
imental data is presented for r/R = 0.69
NOAA FRY-40 Hull
The objective of the NOAA FRY-40 Howfield study
is to examine the fluid behavior in the vicinity of the
propeller appendage. The propeller itself was not used
in the experiment, the purpose of which was to ob-
tain nominal propeller plane data. Experimental data
is available for several circles defined around the pro-
peller appendage; this data is compared to the computed
results. The fishing vessel case utilizes approximately
937,000 points and 5.2M tetrahedra for the entire do-
main with a sublayer resolution of y+ = 2 - 3 on the
hull; the Reynolds number is 7.4 million based on the
body length. A tetrahedral grid is used exclusively for
the fishing vessel case; the surface grid, as well as a lo-
cation at which data is available, is shown in Figure 6.
A sample of computed vs. experimental results for
the fishing vessel case are shown in Figure 7 for a disk
coinciding with the intended location of the propul-
sor; for the computed results, the one equation Spalart-
Allmaras turbulence model is utilized. As shown in Fig-
ure 7, good agreement exists between computed and ex-
perimental results. A distinct velocity defect is seen be-
low the keel (0 = 0°) and behind the prop appendage
(0 = 180°~. Also, the trends in the distributions of Vr
and Vat indicate that the How in the vicinity of the disk is
directed slightly upwards. Agreement between compu-
tations and experiment improves as r/R increases, pre-
sumably due to smoothing of high gradients in the How
very near the appendage surface. A notable exception is
the wake from the centerboard, which is clearly visible
in the experimental results but is not predicted by the
computation at large r/R.
A second set of velocity profile predictions are given
using the two equation q—w turbulence model in Figure
l oo ?~' - ~
O 5 0 r
1 1
1.00 r
> 1 r/R = 0.690 1 0.75
> 0.25 ~ ~
000~ 050
Vx, computed > 0.25
Vr, computed
—0.25 ~ ~ Vt, computed >
o Vx, experimental
Vr, experimental ,
~ Vt, experimental 0.00
-0.50
o
90 1 80
circumferential angle (degrees)
270 360
Figure 7: Velocity profiles for the fishing vessel case at
r/R = 0.69; Spalart-Allmaras turbulence model
8. No significant differences are noted by comparing
the q—w solution with the Spalart-Allmaras turbulence
model solution, (Figure 7), except that a stronger wake
is predicted from the keel by the one equation model as
r/R increases (thus indicating that less eddy viscosity is
present in this vicinity). So, the solution itself seems to
be relatively insensitive to the turbulence model used.
SUBOFF Model
For application to submarines, the SUBOFF model
is presented as a candidate test case. The SUBOFF case
is based on the submarine model given in t361 with How
measurements taken from t371 and t381. The configura-
tions tested here are the nonappended SUBOFF hull and
a SUBOFF hull with four stern appendages.
The Reynolds number is 14 million based on the ref-
erence length of 13.9792 feet (the body length of the hull
is 14.2917 feet). The multielement unstructured mesh
consists of 1.2M nodes, 1.7M prisms, and 2.3M tetrahe-
dra. The spacing of the first mesh point from the body
is 7.2x10-7, which leads to a y+ distribution of less
than 0.5 over the majority of the hull body (thus indi-
cating good viscous sublayer resolution). An overview
of the SUBOFF volume grid in the vicinity of the stern
appendages is given in Figure 9; this figure illustrates
the high aspect ratio prismatic boundary layer elements
as well as the large amount of grid point clustering in
the boundary layer. A typical distribution of y+ on the
submarine body is shown in Figure 10.
The data given in t371 primarily consists of inte-
grated force data given a certain model configuration
and submarine attitude. The axial force, normal force,
and pitching moment are computed and compared to
the corresponding experimental data in Figures 11 -
4:
-0.25
-0.50
o
J
~0
I r/R = 0.690
'= = ~ I:
90 1 80
circumferential angle (degrees)
—Vx, computed
—Vr, computed
—Vt, computed
0 Vx, experimental
Vr, experimental
Vt, experimental
270 360
Figure 8: Velocity profiles for the fishing vessel case at
r/R = 0.69; q—w turbulence model
Figure 9: Volume grid in the vicinity of the stern ap-
pendages for the SUBOFF model
I ~ T ~ r
t ~
- - t - t
~ ~ t
0 02 04 06 08 ~
Figme 10: Computedveluesof;q ondheSUBOFFhull
et ten deg ees mgle of ettack
e
8
e
13 Ag eement is excellent for the axiel forces, normel
forces, md pitching moments et ell mgles of etteck, md
for b odh Spelart -A llmares md q—m turbulence m o de ls
Since the experimental deh itself is nonsymmehic (fhe
dkeg forces for AOA = -10 should equal fhe dkeg forces
for AOA = 10, for exemple), no fimm conclusion mey be
dkewn conceming fhe more fevoreble turbulence model
to select for the SUBOFF simoktions
Experimental deh [38] is elso a~ihtle for the skin
friction coefificient on the SUBOFF nom~ppff~ded hull
For fhis cese, two simohtions are giw~n: one et model
scele (1~ = 1I million, 730,000 nodes) md one et full
scele (/6 = 1.2 billion, IM nodes) 7he q—m tmbu-
lence model is utili:D:d in bodh cases As shown in Figme
14, eg eement betweff~ experiment md computation is
excellent on the eft part of fhe hull (where meesurements
ReRb dab
~S~b~; Onnsm~e
~:
Vt
Figme 12: Nommel force coefificient for fhe SUBOFF
model
elDeRi dab 1
| Vs omem ~ode ~ r ~t t
:-* r
~--: t
. . . ~
Figme 13: Pitchingmomentcoefificientforfhe SUBOFF
model
0 008 r
0 006
0.0041~
'\
0.002 1~- -
11
1
0.000 L
-0.002
Model Scale (Cf. m)
—-— Full Scale (Off)
—- - - 0.527/Cff
o Experiment
____ __
o\
~ _ _ _ _ ~ ~ \
= ~ \\
\\`
N \
\~\
0.2
0.4 06 08 1
x/L
Figure 14: Comparison of full scale and model scale
computations for the nonappended SUBOFF hull
are available). Since the measurements were (obviously)
taken at model scale, the numerical results for the full
scale case have been subjected to a Reynolds number
scaling t391 before comparison to the experimental data.
Like the measurements, the computed skin friction co-
efficients display a definite peak where the How accel-
erates (hull neck-down point, C/L ~ 0.77), and a de-
creasing Of as the How slows between C/L = 0.8 to
C/L = .95. The secondary peak at C/L = .98 is due to
a small How acceleration over the shoulder of the after-
body cap. The capability to robustly simulate full scale
Reynolds numbers is considered a key feature of the nu-
merical algorithm.
DTMB Model 5415 Hull
The DTMB 5415 hull is presented here in nonap-
pended and fully appended form (with dual propulsory.
The primary purpose of the nonappended 5415 hull sim-
ulation is to examine nominal wake How patterns. The
fully appended hull with propulsors is intended to high-
light the capability of the overall solution methodology
to model extremely complex geometries coupled with
complex unsteady Howfields.
Nominal Wake Calculations
Accurate prediction of the nominal wake is a key
item for propulsor design. Here, a computation is
presented for a DTMB 5415 nonappended hull at a
Reynolds number of 12 million. The grid consists of
2.5M nodes, 2.8M prisms, and 5.9M tetrahedral ele-
ments. A rigid lid is used for the waterline, and a sym-
metry plane boundary condition runs lengthwise down
the center of the hull model. Normal boundary spacing
is set such that the nominal y+ value on viscous sur-
faces is 0.5. The q—w turbulence model is used for the
nominal wake calculation.
Figure 15: Nominal wake calculation for DTMB 5415
nonappended hull compared to experimental measure-
ments
Axial velocity data is given in Figure 15 at the
propulsor plane of AL = 0.935 t401. The overalltrends
in the experimental data have been captured by the un-
structured computations. The primary difference is the
amount of thickening in the boundary layer along the
centerline of the hull; the computations seem to over-
predict this phenomenon. A probable cause of this dis-
agreement is that the mesh used in the nominal wake
calculation is insufficiently resolved to adequately cap-
ture the vortices generated by the bulbous bow; hence,
the influence of the vortex structure is not reHected in
the computed results.
Fully Appended with Rotating Propulsors
The DTMB 5415 model with appended shafts,
struts, rudders, and propulsors is presented as a
demonstration of capability of the current unstructured
methodology to 1) model complex geometries, 2) gen-
erate high-quality viscous grids around these complex
geometries, and 3) perform accurate unsteady solutions
on these complicated meshes.
The appended DTMB 5415 simulation is run with a
Reynolds number of 12.7 million and with the Spalart-
Allmaras turbulence model for closure. The grid con-
sists of 2.15M points, 3M prisms, and 3.5M tetrahedra.
A symmetry plane boundary condition is used such that
only half of physical space is simulated, and a rigid lid
is used for the waterline. The time step selected was
1.38x10-4, which corresponds to a prop rotation of one
degree per time step with a nondimensionalized rota-
tional speed of 126.59. Local time stepping was utilized
during the first part of the simulation to quickly estab-
lish wake formations, and minimum time stepping was
then performed for 3.5 prop revolutions to establish pe-
riodicity. Five Newton iterations and seven linear subit-
erations were performed each time step. The rotation of
the propulsors is handled via local grid regeneration.
Figure 16: Surface pressure for the aft end of the fully
appended DTMB 5415 hull with twin propulsors
The pressure distribution on the hull and propulsors
is shown in Figure 16. The surface pressures indicate
the complexity of the Howfield, and the expected trends
are apparent on the struts, shafts, rudders, and hullform.
Further, the effect of the prop wash on the rudders is
visible in the form of a strong low pressure region on
the outboard side of the rudder and a relatively benign
pressure distribution on the inboard side.
Velocity data from the computations are compared
to experimental measurements t411 in Figure 17. In this
figure, contours of axial velocity are shown with veloc-
ity vectors shown for secondary motion. The compu-
tations have closely captured the trends reHected in the
experiment, with the notable exception of the nominal
wake (demonstrated in the previous section). The reason
for the absence of this wake is insufficient grid resolu-
tion behind the keel, such that the convected wake struc-
ture is dissipated before it reaches the propulsor plane.
Predictions of the prop wash, however, are accurate to
the experimental results. Slight differences may be ob-
served on the inboard side of the props, where the effect
of the keel wake allows for easier entrainment of the
How aft of the propulsors.
CONCLUSIONS
A parallel unstructured solution algorithm capable
of time-accurate, high Reynolds number complex geom-
etry simulations has been presented and demonstrated
on several hydrodynamic cases of interest. It is also
demonstrated that rotating propulsors may be simu-
lated effectively in an unstructured environment. Fu-
ture work involves the incorporation of a nonlinear free
surface capability into the unstructured solver such that
propulsor/hull/free-surface interactions may be modeled
and studied in detail. In addition, an effort is cur-
o.o~
i
-0.01
-0.02
-0.03
-0.04
Unstructured Solution Experimental
-11-~ ~~ 11
~~;~ ~-~ 11—1~1:
-0.06 -0.04 -0.02 0 0.02 0.04 0.0~;
ALL
Figure 17: Comparisons of unstructured computation
to experimental data for fully appended 5415 hull with
propulsors
rently underway to incorporate general surface motion
into the unstructured algorithm. This capability allows
prescribed movement of the control surfaces of a given
body to determine maneuvering characteristics.
ACKNOWLEDGEMENTS
This research was sponsored by the Office of Naval
Research under grant number N00014-99-1-0751 and
includes work monitored by Dr. Edwin Rood and Dr.
Patrick Purtell. This support is gratefully acknowledged.
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