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OCR for page 423
Validation of High Reynolds Number, Unsteady Multi-Phase
CFD Modeling for Naval Applications
J. Lindau R. Kunz D. Boger D. Strnebring H. Gibeling
(Penn State Applied Research Laboratory, University Park, Pennsylvania 16802)
Abstract
Un teddy, high Rey old number validation
cases for c multi-phase CFD analysis tool have been
pu sued Tbe tool, designated UNCLE-M has c wide
r mge of applicability including flows of naval
relevance Tbis in ludes supercc itating Ed cavitating
flows, bubbly flows, Ed water entry flows Tbu far the
tool has been applied to c variety of co figurations
A isymmehic sheet cavity flow-fields have been
modeled in putic jar, m crempt to validate file
unsteady r liability of UNCLE M wibb consideration of
the effect of cavitati m number, Rey olds number Ed
tubulen model has been mad Analy is of the
modeled Em teddy fl w field is also mad Ed
con lusions regarding the c uses of su cess Ed
sho tcom logs m the computational re mlts are d awn
Introduction
Tbe ctility to properly model unsteady
multiphase flows is of g eat impo tance, particularly in
naval applications Cavitation may occu in submerged
high peed vehicles es well es rotating mcchin y.
nozzles, Ed mmmerou of her vend s Traditionally,
ca anon on has bad n dative implications associated with
damage Ed or noise However, for high speed
submerged vehicles, file red ~ tion in d cg associated
with c natural or ventilated cavity has g eat pote ticl
ben ft. Yet, cavitation mod leg remain c difficult
task, Ed only recently have full Rey olds-avemged,
tb ee-dimensional, multi-phase, Navier-Stokes tools
reached the level of utility Nat fLey might be applied
for engine enog p moses
UNCLE-M (Klutz, 19990,11)) is c fully
implicit, pa -conditi med. multi-phase, 3-D, fully
gen mlized multiblock, parallel, Rey olds~eraged
Navier-Stokes solver Tbe code was initially evolved
from c version of the single-phase UNCLE code
developed et Mississippi State University (Taylor,
1995), Ed hr. undergone sig fficmt flubber
development UNCLE-M in omorctes mixture volume
Ed constituent volume faction trmsport/gen ration for
liquid, condensable vapor Ed non-condensable gas
fields Mixture momentum Ed tubulen e scalar
equations are also solved Flu limiting has been
applied to file in iscid flu temms based m file local
slope of file solution volume Faction As c r suit, high-
1
order accurate solutions containi g crisp, physically
reasonable mte faces et tb cavity boundary may be
obtain d wibb minimal nonphysical oscillations
Non equilibrium mass transfer mod leg is
employed to ccptme liquid Ed vapor physic excb mge
Tbe code c m b male buoyancy effects Ed the presen e/
interaction of condensable Ed non-condemctle field.
Tbis level of modeling complexity represents file tate-
of-the art in CFD crurlysis of ca vitation Tbe resh ictions
in r mge of applicability associated with in iscid flow,
slender body theo y Ed of her simplffyi g assumptions
are not pa sent In particohr, the code m plushly
add ens the physics associated wibb high-speed
manes rs, body-cc ity interactions Ed viscous effects
such as flow separation
Tbe principal interest here is in modeling high
Rey olds number, umstecdY flow cutout bodies wibb
rummg cavities These cc ities me presumed to be
sheet cavities amenable to c h mogen ous approach
other words, it is presumed flat file non quibbrmm
dynamic for es of bubbles are of n gligible mug it de
In the present work, file effect of surface tension is not
in crporcted, sin e inte face cu vatmes are very mall
for the co figurations considered Tbis cssumpti m is
supported by model results of sheet cavitati m with c
full two-fluid approach (Grogger Ed Akjbegovic,
1 993)
In pa vious work Mum, 1 99hT!), file fidelity
of UNCLE-M has been demonstrated for steady state
fluid flows However, du to the reenh mt jet, cavity
pinching, Ed of her effects of turbulent separated flow,
m iti-phase flows of naval importance me generally
unsteady ~ the work pa sented her, UNCLE-M will
be applied to several co figurations of naval relevance
Each of these co Migrations pa sents m experimentally
documented, unsteady fluid dy amic test case Model
remits will be presented for several ballistic, cavitatcr
geometries Bobb file steady (averag d) Ed unsteady
(time domain Ed pectn 1) behavior of the flow will be
presented Ed compared wibb data h addition,
intere img urn he tdy mmmerical results will be presented
in a field form for comparison with photon sphic dhtn
By comparison of file m merical Ed me tsured results,
the nlhbility of file umsteldy capabilities of file code
m to he understood
OCR for page 424
Symbols:
Cl,C2
Cdest. Cprod
Cp
cd
D
dm
f
gi
k
L
. i
Prn ~Pr
p
ReD
Sh
tu bule pe model const mts
mass hemsfer model comstemts
pressure coefficient
d cg coefficient
body diemeter
bubble diemeter
cyclingfiequ py Hz)
g~avity vector
tmbuient kmetic energy
bubble le 3th
mass t msfer rctes
tmbulent kmetic energy produption
tu bulent P mdtl numbers for k emd e
press lre
Rey olds number beped on body diemeter
Shonh~lfrequ py (fD)/U=
e P ienf h clong co figmation (clso secomds)
t, t=, At physiccl time, meem flow time sccle, time step
velocity mcg itude
Cc tesi m velocity components
Cc tesi m coordinctes
U
Uj
Y+
D
p
U
p
Subscripts, Superscripts:
D
t
dimensionless wall di te pe (p yU~)/~
volume f ction, emgle of ctt~ck
preconditionmg pe emeter
pseudo-time
tu bulence dissipation rcte
molecohr viscosity
density
cc itstionmmber( = P= Pv
1/2p,U
body diemeter
liquid
ml tme
noncond nsable gas
tmbulff~t
condemctle vapor
free t~eem valu
Physical Model
he phy iccl model equ~tions solved he~e
have been decribed previpmsly ~u d 1999 0,11)) he
basis of the model is the i pomp~ ble multiphase
Rey olds Avereged Ne ier Stokes Equstipms in c
homog neous form Each phcse is treeted es c new
species emd requi es the mclusion of c sepe cte
contimmity equstion Ihree species, ~epresenti g c
liquid, c condensable vapor, emd c no pond nsable gas,
e e i pluded Miss hem fer betw en fhe liquid emd
vapor phcses is cchieved f ough c dffferenticl model
ther resee chers have cpplied simile models with c
smgle species cpprocch How ver, the multiple species
model of multiphase flow is p~esented es c mme
flexible physiccl cpprocch A high Rey olds m mber
fomm of twocqu~tion models with stemde d wall
fu ptions provides tu buie pe closu e
he govermog dffferenticl equ~tions, cept m
Cc tesiem tensor form e e given es Equ~tion (1):
( 2~a~+a~j
(P~Uj)+d p~Uj)+a] (p~UjUJ
a~i ~ ~i jap+a~i+ a
i ~p 2)a~ a~ a~p
at +( p2~ +a~ e+d (~eU) = 0
( )( )
, Pi PV
= dPx +a~j(~mytW)+p~g
(~ U ~ = (m +m ) (1)
Where ml tue densit emdtubulentvicosity
have been deftned m Equ~hon (2)
pm = p,~ + Pv4 + P~d~d
p~C~k
F~' e
(2)
In fhe present work, fhe density of ecch
constituent is teken es comtemt Equ~tion (1) rep~esents
the conservation of mixtme volume, ml tme
momentum, liquid phcse plume f ction emd nom
condensable gas volume f ction, respectively Physiccl
time derivatives e e i pluded for umstecdy
computations he formuletion mcorpomtes p~e-
conditioned pseudo-time-derivatives (a/8s terms),
defmed by pe emeter p, which provide fe orable
converge pe che cteri tics for stecdy state emd
umstecdy compu ctions, es discussed fu ther below
he formation emd colhpse of c cc ity is
modeled es c phase t msfommation Debiled modeling
of fhis process requi es k owledge of fhe
themmodynamic behavior of the fluid nee c phepe
t msition pomt emd the fommation of interf ces
Simplffied models ere presented here, ~esulting m fhe
use of empi iccl f ctors Given es Equ~tion (3), two
sepe cte models e e used to descobe fhe t msfommation
of liquid to vapor emd fhe t msfommation of vapor back
to liquid For hemsformation of liquid to vapor, i is
modeled es being proportiom~l to fhe produpt of fhe
hqmd volume fi ctmn emd fhe d'fferffce between fhe
2
OCR for page 425
computatlord~l cell pr ssme md the vapor pressu e
This model is simibr to fhe or used by Merkle et cl
(1998) for bodh evaporction md condensatior For
trmsformation of vapor to liquid, c simplffied form of
the Girdbuglmd~u potff~tial is used for fhe mas
trmsfer rcte ~
~ =
cdestpv~iMIY[O P [v]
( Pi =)=
+ CpVodPv~i (} ~i)
Cde~t mdCrOdare mpiricclconstmts Forcll
work preented here, Cd~st = CprOd = 105 Both mass
trmsfer rctes are nom-dimenriord~lized with r spect to c
me m flow time sccle
In fhis work, c high Rey olds mmber two-
equation tmbulerpe model wifh st mdard wall f mctions
hcs been implemfftted to provide tubulerpe closue
Eifher the k-e or RNG k-e (Orszag et cl 1993) model
cc represented m Equation (4):
i(P~t)+~(P~tU,) = a~,(Pvtta—X,)+P ps
i ~ a~j ~ J a~j( t~a~,) I ~ t)
(4)
As with velocity, th tmbulffce pclars are
interpr ted dP being mixtme qu mtities
Numerical Method
The baselme mmmericcl method hcs been
evolved from the work of Tcylor md his coworkers et
Mississippi State University (Tcylor et cl (1995), for
e dmple) Prim~tive varictle mterpolmt t pe Roe flu
d'ffer rce spl~rmg ~s ued for spat~cl d' pret~zatmn A
implicit procedme is cdopted wifh ir i pid md vi cour
flu Jccobi ms cpproximated m mericclly A block
symmehic G mss-Seidel itemtion is employed to solve
the cppr oxim ate Newt on ystem et eah t ime step
Th multi-phase extemion of the code retams
the e umderlymg mmmencs but mpo porctes two
cdditiorul volume fraption constit ent trmsport
equations Du ing flu formulation, c Jdmeron-style
(Jdmeron 1981) flux limiter bced on liq id volume
frdption is cpplied to th primitive mte pol mts A nom
diagord~l pseudo -tim e- deflvat ive prec onditi onmg matrix
~s clso employed Wh~le fhe t~me denvat~ve term
mishes from the mixtue contmnity equation es fhe
limit of mcompr ss~ble con titu nt phcses is
cpprpahed, the effect of pr conditionmg is to redupe
the daocidted stiffLess This preconditior r gives rise
to c system with w 11-conditioned eigern~lu s which
cc mdependent of densdy mtm md loccl volume
frd ption This sy tem is w 11 suited to high density rctio,
phcse-separcted two-phcse flpws, such es the cavitatmg
sy tems of mter st her
A t mporally secomd-ord r cocu cte dpal-time
scheme was impleme ted for physiccl time integ ction
At eah time step, the tmbulerpe trmsport equations
are solved mbsequ nt to solution of fhe me m flow
equations The multiblock code is inshumented wifh
MPI for pmallel execution based on domcm
decomposition Du ing umstecdy time mteg ction, to
obtarn remits presented her message pcsstog WdP
') cpplied dfter eah symmetric G mss-Seidel sw ep Each
imer iterdte ir plved twenty symmehic Gmss-Seidel
sw eps, md eah time tep mvolved ffteen imer
iterctions This procedu e was s fficient to relialy
red p the umstecdy residual by et lecst two orders of
mcgmitude However, c pcse by pcse e cmird~tion lik Iy
could have redu d fhe expended computatlord~l effort
yieldi g remits similar in solutiom fidelity Fu th r
detcils on fhe m mericcl method md code are available
inKu det cl (199900
Results
A isymmehic sheet pavity flow-felds have
been mod led in putic dar, m crempt to validate fhe
umstecdy relictility of c m dtiphcse, computatlord~l fluid
dyrdmms tool w~fh com~demtmn of th dffectr
R y olds number md tu bulerpe model hdP beff~ mcde
Stecdy, averag, mecsurements of rlevmt cavitation
pardmeters tor the shapes chosen have been
documentedbyRouse mdMcNow (1949) Stinebring
et cl (1983) documented the um tecdy cycling behavip
of evercl axi mmetric cavitators Their report
irpluded r suits for bodh ventilated md natu cl
cavitation The umstecdy pe form mpe of c 45° (22 5° in
profle from centerlire to outer edge) conical,
hemisphericcl, ad Occhber ogival pavitatms et c rage
of cavitatipm m mbers w re documented Aifhough
UNCLE-M hcs the ccpability to model ventilated
cavitation Ku d 1999([)), only rd~tmcl cavitation
re mlts have been irpluded here it should be noted that
the results of Rouse md M Now (1948) indicated that
for the pavitatm t pes md flows et or ctove the r mge
of experimental R y olds m mbers r ported md
ir stigated here, the flow should be tu bulent over c
sig if icmt portion of the forebody Therefore, fp
smgle phcseflow. particohrlyfor geometricclly smooth
shapes, this should ser to avoid the w 11 k ow
chcotic, criticcl kmmar separction md tr msition
regime The m merical r suits employ c fully tu bulent
model
R suits pre ented her are given in fhe model
computatiord~l system (S~ umits For cll computations,
the free str dm velocity was set to I (m/s), the liqmid
density WdP 1 OOO kg m3), md the vapor density was I
kgm3) For most computations, fhe liquid vicosity
was then set equal to 10 3 P~s), md that of the gas
phcses was set to 10 5 Pc-s) Then the body diameter
was chosen to ahieve fhe desi ed model R y olds
m mber In fhe case of the hemisphericcl forebody rm
et c body didmeter based R y olds m mber of 1 36xl 07,
the liquid kir matic viscosity was then set equal to 10 5
3
OCR for page 426
(Pa-s), and that of the gas phases was set to 10-7 (Pa-s).
The model body diameter for this case was thus, 0.136
(m). Prior to initiating unsteady computations, for
purposes of computational expediency, a steady state,
At = no, integration was carried out. At the completion
of this integration, it was possible to determine if the
model solution was physically unsteady. In general,
physically unsteady conditions were indicated by
marginally convergent, flat-lined steady-state residual
histories, themselves containing large amounts of
unsteadiness.
.. ~
—i s
—1 · _
~ - 1 ~ _
Figure 1: Zero caliber ogive in water tunnel at
Re(D)=2.9xlO5, 0=0.35 (approximate) (Stinebring,
1976).
A photograph of a O-caliber axisymmetric
cavitator operating at conditions similar to those
modeled here is given in Figure 1 (Stinebring 1976)
Figure 2 contains a series of snapshots of the volume
fraction field from an unsteady model computation of
flow over a blunt cavitator. Here the Reynolds number
(based on diameter) was 1.46x105 and the cavitation
number was 0.3. The time history for this case is given
in model seconds, and at t=O, unsteady integration was
initiated after obtaining a steady-state, At = no, initial
condition. Thus it is expected that there was some start-
up transient associated with initialization from an
artificially maintained set of conditions. For the volume
fraction contours, dark blue indicates vapor, a liquid
volume fraction of less than 0.005, and bright red
indicates liquid, a volume fraction of one. Some
significant numerical integration time parameters for
this case are the body diameter to free stream velocity
ratio, D/Uo<' = 0.146 seconds, and the physical
integration step size, At = 0.001 seconds.
This result is presented over an approximate
model cycle. The figure also includes the corresponding
time history of drag coefficient. Note that the spikes in
drag near t=37.725 and t=38.925 seconds correspond to
reductions in the relative amount of vapor near the
sharp leading edge. This marks the progress of a bulk
volume of liquid from the closure region to the forward
end of the cavity as part of the reentrant jet process.
Although far from regular, these spikes also delineate
the approximate model cycle. This picture serves to
illustrate the basic phenomenon of natural sheet
cavitation as it is best captured by UNCLE-M. This
result is notable for the spatial and temporally irregular
nature of the computed flow field. Even after
significant integration effort, a clearly periodic result
1.~:
v
~ .
1.4
1.1
t (s)
Figure 2: Modeled flow over a O-caliber ogive. Liquid
volume fraction contours and corresponding drag
history. UNCLE-M result. o=0.3. ReD=1.46xlO5.
4
OCR for page 427
had not emerged. Thus, to deduce the dominant
frequency with some confidence, it was necessary to
apply ensemble averaging.
An examination of the flow pattern captured
suggests qualitative validity. Note, in Figure 2. that
over a significant portion of the sequence, the leading,
or formative, edge of the cavity sits slightly
downstream from and not attached to the sharp corner.
In their experiments, Rouse and McNown (1948)
observed this phenomenon. They suggested that this
delay in cavity formation was due to the tight
separation eddy which forms immediately downstream
of the corner and, hence, locally increases the pressure.
The corresponding evolution of cavitation further
downstream, at the separation interface, was proposed
to be due to tiny vortices. These vortices, after some
time, subsequently initiate the cavity. Figure 3 shows a
single frame at t=37.8 seconds from the same model
calculation (as shown in Figure 24. Here, to clarify what
is captured, the volume fraction contours have been
enhanced with illustrative streamlines. Note that these
are streamlines drawn from a frozen time slice.
Nonetheless, if all of the details envisioned by Rouse
and McNown were present, the streamlines should ,
indicate smaller/tighter vertical flows. The current level V
of modeling was unable to capture small vertical
structures in the flow. However, the overall
computation was apparently able to capture the gross
affects of these phenomena and reproduce a delayed
cavity. In fact from examination of the cavity cycle
evolution shown in Figure 2, and the streamlines shown
in the snapshot, it appears that gross unsteadiness is
driven by a combination of a reentrant jet and some
type of cavity pinching (Brennan 19924. The pinching
process is particularly well demonstrated in Figure 2
from t=38.125 to 38.325 seconds. However, rather than
complete division and convection into the free stream,
it should be noted that, in later frames of Figure 2, the
pinched portion of the cavity appears to rejoin the main
cavity region.
Figure 3: Snapshot of modeled flow over a O-caliber
ogive. Liquid volume fraction contours and selected
streamlines. UNCLE-M result. o=0.3. ReD=1.46xlO5.
The low frequency mode apparent in most of
the experimental O-caliber results appears to have been
captured at the lowest cavitation number (~=0.3), as
shown in Figure 2, and is evidenced in the test
photograph (Figure 14. In Figure 4, the drag coefficient
history for a 40 model second interval from the same
computation as in Figure 2 is shown. Here, a clear
picture of the persistence, over a long integration time,
of the irregular flow behavior is documented. At higher
cavitation numbers, the current set of O-caliber cavitator
results indicate a more regular periodic motion. This is
contrary to the experimental data. However, as Figure 3
indicates, the ability to capture this motion at any
cavitation number may not necessarily require the
explicit capture of the finer flow details of the vertical
flow structure. This is encouraging and suggests that
with increased computational effort, without altering
the current physical model, the representation of this
phenomenon could be improved over a greater range of
cavitation numbers.
1 f
1.'
t (s)
40
Figure 4: Model time record of drag coefficient for flow
over a O-caliber ogive at ReD=1.46xlO5 and o=0.3. In
model units, D/U~ = 0.146 (s), physical time step,
At = 0.001 (s).
Figure 5 presents the spectral content of the
result given in Figure 4. This power spectral density
plot is based on four averaged Hanning windowed data
blocks of the time domain result. To eliminate the start-
up transient effect, the record was truncated, starting at
t=10 seconds and, to tighten the resulting confidence
intervals, more time domain results, after t=40 seconds
were included. As is typical of highly nonlinear
sequences, the experience of this unsteady time
integration demonstrated that, additional time records
merely enrich the power spectral density function.
However, the additional records do serve to improve
the confidence intervals, and, therefore, add reliability
to the numerical convergence process. The model result
used, was, as indicated by the confidence intervals,
sufficient for a comparison to experimental, unsteady
results.
Figure 6 contains a time record of drag
coefficient during modeled flow over a O-caliber ogive
at a Reynolds number of 1.46x105 and cavitation
number of 0.35. The Strouhal frequency based on this
result is 0.0909. Here it is apparent that the
computational modeling was incapable of reproducing
5
OCR for page 428
a'
3
CAL, 2.5
¢ 2
V
4
35
. .
1.5 ~
o
O ~
~ 4
f (Hz)
6 8 10
Figure 5: UNCLE-M result. O-caliber ogive at
ReD=1.46xlO5 and o=0.3. Power spectral density
function with 50% confidence intervals shown.
what should have been a lower frequency result with
flow around the forebody dominated by a more
irregular cavity. It is supposed that the correct result, in
comparison with the experimental data in the Strouhal
frequency plot (Figure 22) would have been similar in
nature to the results presented for a cavitation number
of 0.30 in Figure 4. In addition to lacking the rich
frequency content of the result for lower cavitation
numbers, it appears that the amplitude of the
unsteadiness present is an order of magnitude lower.
1.171r
1.1705
1.1695
~ ~9
1.1685
Figure 6: UNCLE-M result. Time record of drag
coefficient for flow over a O-caliber ogive at
ReD=1.46xlO5 and o=0.35. In model units,
D/Uo<' = 0.146 (s), physical time step, At = 0.001 (s).
Figure 7 contains a series of snapshots from
the unsteady model computation of a hemispherical
cavitator at a Reynolds number (based on diameter) of
1.36x105 and a cavitation number of 0.2. This result is
presented over a period slightly longer than the
approximate model cycle. In this case the model
Strouhal frequency is 0.0326. There are ten frames
presented, and the first (or last) nine of those ten
constitute an approximate model cycle. The drag
history trace in Figure 8 demonstrates how, relative to
the modeled flow over the blunt forebody, the pattern of
flow over the hemispherical forebody is regular and
periodic. This is consistent with experimental
observations made (for example) by Rouse and
Figure 7: Liquid volume fraction contours. Modeled
flow over a hemispherical forebody and cylinder.
UNCLE-M result. o=0.2, Re(D)=1.36xlO5.
Mcnown (19484. Note the evolution of flow shown in
Figure 7 as it compares to the drag history shown in
Figure 8. As would be expected, the large spike in drag
corresponds to the minimum in vapor shown near the
modeled t=1.6 seconds.
The next three figures demonstrate the
expected and captured dependence of Strouhal
frequency on cavitation number. Here the trend of
increasing cycling frequency with cavitation number
during flow over a hemispherical forebody is
reproduced. The result has been demonstrated at a
Reynolds number of 1.36x105. This Reynolds number
was intended to scale the problem properly with the
data available. Here, the magnitude of the drag and the
amplitude of the unsteadiness may be examined. Figure
9 contains a time record of drag coefficient during
modeled flow over a hemispherical forebody and
6
OCR for page 429
lo is
t (s)
1 1
30 3s
Figme 8 Unsteadyd agcoefficient Flow over6
hemispheric6 I forebody 6md cylmder UNCLE-M res flt
3=0 2, R ~)=1 36xl 05 in model mits,
D/U== 0136(s),physic61timestep,At = 0001(s)
cylmder 6t 6 Rey olds mmber of 1 36xl05 6md
c6 itation n mber of 0 25 The Stro~l frequency
based on this resflt is 0 0484 Figme 10 contams 6
simil6 time record of d 6g coefficient durmg modeled
flow over 6 hemispheric61 forebody 6md cylmd r 6t 6 " ~'
R y olds n mber of 1 36xl 05 6md cavibtion n mber of
0 30 The Stro~l fiequency based on 6his resflt is
00622 Figure 11 contains 6 time record of dag 0465
coefficient durmg modeled flow over 6 hemispheric61
forebody 6md cylinder rt 6 Rey olds m mber of
1 36xl05 6md c6 itation mmber of 0 35 The Sho~l
f~equency b6 sed on 6his re mlt is 0 0933 in 6 ddition, 6he
higher 3, higher fiequency ~esflts co tain sm611er
c6 ities in 6hese sit rti ms, c6 ities d ive the over611
msteadiness of the flow, 6md 6he problem of s fficient
g id pomts to defme 6m msteady c6 ity becomes
6pp6 ent Th3s, by p3shing the limits of ~easom~ble
discretization, 6he limits of effective modeling also 6re
te ted
Figme 12 contams 6 time lecord of dag
coefficient durmg modeled flow over 6 hemispheric61
forebody 6md cylinder rt 6 Rey olds m mber of
1 36x105 6md c6 itation n mber of 0 3 Th Stro~l
f~equency based on 6his resflt is 00614 Figme 13
contains 6 time record of d rg coefhcient during
modeled flow over 6 hemispheric61 forebody 6md
cylmder 6t 6 Rey olds mmber of 1 36xl07 6md
c6 itation n mber of 0 3 The Sh o~l f~equency based
on this remit is 0 133 Here, 6he stamd6 d t~end of
inmeased turbflent cycle fiequency with increased
R y olds m mber mpe6 s to have been p~esented
Figme 14 contams 6 time lecord of dag
coefficient durmg modeled flow over 6 conic61
forebody 6md cylinder rt 6 Rey olds m mber of
1 36xl05 6md c6 itation n mber of 02 Th Stro~l
f~equency b6 sed on 6his res flt is 0 0383 As 6mticipa ted,
due to the expected stability of c6 ities 6bo 3t this
7
04 _
044 ll
04 1e
043
043
Figme 9: UNCLE-M res flt Time record of d 6g
coefficient for flow over 6 hemispherical forebody 6md
cylmderatR D=1 36xlO56md 3=025 ~model mits,
D/U~ = 0 136 (s), physic61 time step, At = 0 0025 (s)
0 475 _
20
Figme 10: UNCLE-M res flt Time lecord of d 6g
coefficient for flow over 6 hemispherical forebody 6md
cylmder 6tReD=1 36xl05 6md 3=0 3 Inmodel mits,
D/U~ = 0 136 (s), physic61 time step, At = 0 0025 (s)
shape, 6his model flow e hibited very regmbr cycling
with little 6 dditiom~l sh ong compone ts from second6 Y
modes
Figme 15 contams 6 time record of dag
coefficient durmg modeled flow over 6 hemispheric61
forebody 6md cylinder rt 6 Rey olds m mber of
1 36xl 05 6md c6 ibtion m mber of 0 25 This is 6moth r
UNCLE-Mresflt;how ver,ratherthmthe tamd6 dk
3 mod 1, 6he RNG k-3 tmbflence model has been
6pplied For the h ml pheric61 forebody wi6h
cylmd ic61 6fterbody, when 3smg the stmd6 d k 3
model, to obtain, during 6 complete d al time cycle, 6
reduction m the mstecdy residual of two orders of
magmit3de, it was mfficient to 6pply 6 time step,
At=0 0025 seconds However, wi6h the RNG k 3 m odel,
to obt6 in the same ~eduction in 6he mste6 dy ~esid al, it
OCR for page 430
0 496 _
O AOA
~ 492
Figme 11: NCLE-M ~esult Time ~ecord of d cg
coefficient for flow c er c hemispherical forebody md
cylmder ctReD=1 36xl05 md 5=0 35 Inmodel mits,
D/U~ = 0 136 (s), physiccl time step, At = 0 0025 (s)
Figmel2: NCLE-Mremlt Time~ecordofdag
coefficient for flow c er c hemispherical forebody md
cylmderstRD=136xlO6 md 5=03 Inmodl mits,
D/U~ = 1 36 (s), physiccl time step, At = 0 025 (s)
was nes ssary to m c physiccl time step of 0 001
seconds Note that in this time hi to y hcse, there is c
g est decl of m tecdmess The result mpears far less
cohe~ent th m the st mdard k-7 re mlt given m Figme 9
The Sh o~l fieque sy based on 6his ~esult is 0 1855
Bcsed on 6he mecsured dsh (Stinebring 1983), this
f~eque sy is far too high When cpplied for c higher
cavitation n mber, 5=0 30, the RNG k'based model
cgam requi cd c smeller time tep (O 001 mits) md
predicted c Sh o~l f~eque sy of 0 068 Here 6he value
is nearly th ssme es 6~t predicted by 6he model using
the k~ turbule se mod I Clearly, the t~end based on
these results is i sorrect it cppears that 6he current
implementation of the RNG model hcs yielded ~esults
consistent wi6h the k-7 model st one savitation m mber,
5=0 30, but et c 1ower value, the cycle f~eque sy is far
g ester 6 m the t mdard k-7 modeled ~esult or 6he
measured data it seems probable that wi6h finer time
s &ilO 15 ~o
t (s)
Figme 13: NCLE-M result Time ~ecord of d cg
coefficient for flow c r c hemispherical forebody md
cylmder ctReD=1 36xl07 md 5=0 3 Inmodel mits,
D/U~ = 0 136 (s), physiccl time step, At = 0 0025 (s)
Figme 14: NCLE-M result Time ~ecord of d cg
coefficient for flow over c comccl forebody md cylind r
st R D=1 36xl05 md 5=0 3 Inmodel mits,
D/U~ = 0 136 (s), physiccl time step, At = 0 0025 (s)
md spcse di sretization. the cmrent RNG k-7 model
implementstion wo 5d cchieve remits comparable wi6h
the k~ mod I et cll cavitation n mbers As expected,
the RNG model i sreased 6he owxcll m tecdmess of
the ~esults How ver, the computatiorurl cost of 6he
cunent res 5ts is checdy sig ffic mt, md based on 6he
NCLE-M sohtiom obtamed thns far, md comparison
to experimentcl data, little benefit mpears to be had
from 6he cun ent cpp licati on of the RN G k-7 m o de l
Whe~e mplic~ole, for 6he unstecdy ~esults
presented he~e, the ari6 meticclly avemged re mlts h~ve
been compmed to the ~esults of Rouse md McNswn
(1948) Figme 16 contams c comparison for flow over
the O-caliber savitstor. Figm~e 17 contains c similar
comparison for flow owx c hem isphericcl cc itator, md
Figme 18 contams c similar comparison for flsw over c
8
OCR for page 431
0 44s
n~
0 425
4,
8
t (s)
FigmelS: NCLE-M tNGk-8turbulencemodel
result Time record of d cg coefficient for flow over c
hemisphericcl forebody md cylmder et ReD =1 36xl 05
md 5=025 ~model mits,D/U~ = 0136 (s),physiccl
time step, At = 0 001 (s)
i i\ i _ _ _ oomput diOn, GO 3
~ ~ dd~G03
o ~ 1 i oomp d diOn, GO 4
o ~ T i ~ dd~G04
O Ll i i i
v~o ~ I i i,~.~ i i
O I i W
O ? ~ ~
O~ i i
0 1 4 ~d 6 8 10
Figme 16: Flow owx c Occliber cavitator (s/d~rc
ienf h over dismeter) Averaged m tecdy pressure
computations mdmecsuredGtc Rouse mdMcNown
1 948)
coniccl cavitatcr in ecch of 6hese figmes, 6he overcll
pe formance of 6he cod setms to g nerclly cg ee with
the Gh Clearly es the cavitation m mber is ~educed,
the NCLE-M result tends fur6her fiom 6he mta For
both the m mericcl md experimental ~esults, 6he
a~emge initiction md temmim~tion pomt of 6he cavity
may be deduced fiom 6his figme Accordingly, 6he
ctility of the code to properly model the average cavity
is well ~epresented in 6hese fgmes The a~emged
pe formance over the Occliber G itator cppears better
th m that of eithb of the others The pe fommance over
the coniccl shape is the wor t it is clear th~t 6he
fommation pomt of the average cc ity should be well
defmed m the axisymmehic shapes wi6h dicontmoous
os
oc
,~o.
~ r ~ oomput diOn, G0 1
\ ~ dd~G02
. | oomput diOn, c=0 3
dd~G03
1 '
~ t i i i
o ~ 1 ~ s, i
o ~ . --=~-t-~
o~
04 . i i i
0 1 ~ 3 4 5 6
~d
Figme 17: Flow over c h m isphericcl ca vitator (s/d~rc
ienf h over d smeter) Averaged mstecdy pressure
computations mdmeaeuredGtc Rouse mdMcNown
1 948)
o.
v
o.
Ob
oomput diOn, GO 3
06 ~ dd~G03
0 4 oomput diOn, GO 4
~ ~ dd~G04
O? :~. . i i
_ O ~ ''~'~
O ~ ':,~' ~ ~ -
~ .~iL .
0 ~d ;
Figm~e 18: F!ow owx cconiccl G itator (sM~rc ienf h
overd~smeter) Avemged mstecdypressure
computations mdmeaeuredGtc Rouse mdMcNown
1 948)
Severcl parsmeters of ~elevance m 6he
charcterization of cavitation bubbles inchde body
dismeter, D, bubble ienf h, L, bubble dismetb, dm, md
fomm d cg coefficie t cssocicted wi6h th cavitator, Cd
Some smbig ity is mherent m both the expb imentcl
md computatmrurl dehmhon of the ktter th ee of these
parsmeters Bubble closme location is difficult to
defne due to mstecdmess md ~ts dependence on diter-
body d6smeter (which c m r mge from O [isohted
camtator] to the camtator d~smeter) Acccrdmgly,
bubble length is often, md he~e, tskb es twice 6he
9
OCR for page 432
distance from cavity leading edge to the location of
maximum bubble diameter (see Figure 194. The form
drag coefficient is taken as the pressure drag on an
isolated cavitator shape. For cavitators with afterbodies,
such as here, the pressure contribution to Cd associated
with the back of the cavitator is assumed equal to the
cavity pressure (- Pv) For the model computations, dm
is determined by examining the al = 0.5 contour and
determining its maximum radial location.
In Figure 20, the quantity L/(DC4/2) is plotted
against cavitation number for experimental data sets
assembled by May (19754. Arithmetically averaged
UNCLE-M results are included for ten unsteady
computations made with three cavitator shapes. The
correlation between L/(DC4/2) and ~ has been long
established (see Reichardt (1946), Garabedian (1958),
for example). Despite the significant uncertainties
associated with experimental and computational
evaluation of L and CD, the data and simulations do
correlate well, close to independently of cavitator
shape.
~....................
Figure 19: Definition used to determine bubble length,
L, and diameter, dm.
loot , , ,,,,,, I i
1 0
. . . . . . ... . . . . . . . . . . . . . . .
:::::::::::::
.................... ..
. ~ ::::::::
......................
. .......
.......................
: : : : : :::
1 0°
lo-2 lo-l 10°
UNCLE-M hemisphere
UNCLE-M O-caliber
UNCLE-M cone
data sphere
data stagnation cup
data cone
Figure 20: Dimensionless drag to bubble length
parameter and cavitation number. Flow over
axisymmetric cavitators. Arithmetically averaged,
unsteady UNCLE-M results and data (May 19754.
Another parameter that has been established to
be well correlated with cavitation number is the
fineness ratio, L/dm. May (1975) noted that this
parameter is particularly independent of ambient
pressure, vapor pressure, free stream velocity, and
whether the cavity was filled with vapor or a mixture of
vapor and air. Once again, May assembled a large
quantity of experimental results for this parameter.
Figure 21 contains a comparison of the fineness ratio,
L/dm, for averaged unsteady UNCLE-M computations
and data.
As a blanket observation, the spread of data
between the experiments and computations in Figure 22
appears to be rather large. However, there are several
encouraging items to be reviewed. It is clear that (for a
given cavitation number) the computational results are
bounded by the experimental data, and the proper
trends (rate of change of Strouhal frequency with
cavitation number) are well captured. More insight into
the physical relevance of the data requires examination
of specific results.
2
lol
10°
10
-- O UNCLE-M hemisphere
UNCLE-M O-caliber
~ D UNCLE-M cone
--I + data cone -I
-- X data stagnation cup
~ ~ data sphere
~ + data disk
~
10-2 ~ 10 10°
Figure 21: Cavity fineness ratio and cavitation index.
Flow over axisymmetric cavitators. Arithmetically
averaged, unsteady, UNCLE-M results and data (May
19754.
When run at similar cavitation numbers, the
extremely low frequencies observed in the 0-caliber
ogive testing was not captured by the model. However,
considering only model results at a cavitation number
of 0.3 (see Figure 4), it appears that the observed of
behavior was captured.
Figure 22 contains a large survey of unsteady
computational and experimentally obtained data
(Stinebring 19834. The numerical results in this figure
summarize this validation effort. Here, Strouhal
frequency is shown over a range of cavitation numbers.
Computational results are given for hemispherical, 1/4-
caliber, conical, and 0-caliber forebodies. Unsteady
experimental data is included for the hemispherical,
conical and 0-caliber shapes. Computational results for
the hemisphere, 1/4-caliber and conical forebodies,
were obtained at a Reynolds number based on diameter
10
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OCR for page 440
Representative terms from entire chapter:
rey olds
of 1.36x105. For the O-caliber ogive, computations were
made at a Reynolds number of 1.46x105. In addition,
for the hemisphere, results are included for Reynolds
numbers of 1.36x106 and 1.36x107. The experimental
results included in the figure were obtained at Reynolds
numbers ranging from 3.5x105 to 1.55x106.
0.5
0.4
hemisphere ReD=(0.35-1.55)xlO
O-cal ReD=(0.96-1.46)xlO5
cone ReD=3.9xlO5
model 1/4 calReD=1.36xlO5
model O caliber ReD= 1 .46x105
model hemisphere ReD=1.36xlO5
model hemisphere ReD=1.36xlO
model hemisphere ReD=1.36xlO7
model cone ReD= 1.36x105
0.3
0.2
0.1
_
0 0.
`1'
-
-
· ~,~ & ~
0.2 0.3 0.4 0.5
Figure 22: Axisymmetric running cavitators with
cylindrical afterbodies. Strouhal frequency and
cavitation number. UNCLE-M results (open symbols)
and data reported in Stinebring (19834.
For the hemispherical forebody results, as may
be seen in Figure 22, there is a significant but almost
constant offset between the measured unsteady data and
the modeled results both of which appear to follow a
linear trend over the range presented. An interesting
result occurs in the model data for the hemispherical
forebody with a Reynolds number of 1.36x107
(pentagrams in Figure 224. Here the numerical results
appear to agree quite well with the experimental data
for hemispherical forebodies. The experiments were
taken at an order of magnitude lower Reynolds number,
but the agreement is apparent in both cases where
model results have been obtained. For design purposes,
this may suggest an avenue towards model calibration.
Another result found in the Str versus ~ plot
(Figure 22) is the tendency of the modeled flows to
become steady at higher cavitation numbers. For the 0-
caliber or the conical cavitators, this is the reason model
results are not included for cavitation numbers greater
than 0.4. For the modeled hemisphere, the upper limit
of cavitation number to yield unsteady model results
was found to be Reynolds number dependent. At a
ReD=1.36xlO5, the maximum cavitation number
yielding an unsteady result was on 0.35, at
ReD=1.36xlO6, that number was ~ ~ 0.45, and at
ReD=1.36xlO7, the maximum cavitation number for
unsteady computations was not determined. This result
may indicate a limit of the computational grid applied
to the problems rather than a limit of the level of
physical modeling. In addition, physically in the mode
of unsteadiness present, a transition does occur from
cavity driven to separated, turbulent, but single phase
driven flow.
For the conical forebody, the datum shown in
Figure 22 suggests that the cycling frequency should be
higher, 0.123. It is worth considering that the Reynolds
number of the experimental flow was 3.9x105 and that
the general trend with increasing Reynolds numbers is
to increased frequency. However, based on the standard
level of dependence of Strouhal frequency (see
Schlicting 1979 for example) on Reynolds number for
bluff body flows, it would seem unlikely that the rate of
change in frequency with Reynolds number (at
ReD~ 105) would be as high as three to two. In
addition, compared to shapes with geometrically
smooth surfaces, the nature of unsteady flow over a
conical shape is not expected to be nearly so dependent
on Reynolds number. In the case of a cone, at low
values of cavitation number (i.e. o=0.3), the separation
location, and, hence, the likely forward location of the
cavity, is rarely in question.
A trend that is captured in the model results
but not represented in the experimental data included
here, is the tendency for the Strouhal frequency of a
given cavitator shape to exhibit two distinct flow
regimes. The first regime exists at moderate cavitation
numbers and is indicated by a low Strouhal frequency
where the value of Str will have an apparent linear
dependence on is. The second regime tends toward
much higher cycling frequencies. Here the dependent
Strouhal frequency appears to asymptotically approach
a vertical line with higher cavitation number, just prior
to the complete elimination of the cavity. This is
documented in Stinebring (1983) and demonstrated in
Figure 22 for the modeled hemisphere at
ReD=1.36xlO6. Based on the model results, it appears
that this is characteristic of a change from a flow mode
dominated by a large unsteady cavity to one dominated
by other, single-phase, turbulent, sources of
unsteadiness.
During this investigation, some effort towards
the establishment of temporal and spatial discretization
independence was made. As a requirement of the
model, to accommodate the use of wall functions, for
regions of attached liquid flow, fine-grid near-wall
points were established at locations yielding
lO
(sh wn) flequ n ies, 6here was very similar modal
behavior Unfortmutely only 6he fin -g id models
t nd d to pro ide umstecdy remits Thus time md
spaticl fidelity w re indged mdependently A
demonstmtion of the tecdy-state spaticl convergen Of
the modeled coniccl forebody md cylind iccl dfterbody
is given m Figme 24
m~
V I O
Figme 23: Spectrcl comparison of effect of phy iccl
integ ction time step si:D: on Cd history UNCLE-M
~esult Flow over c hem i phericcl forebody wi6h
cylind iccl sfterbody R D=1 36xl05 5=0 3
o
~ = Medium Gnd
l \
1 \
~ ~ 1~
. \ 1!
I //
I . . I . . I . . I . . I
0 1 ~ 3 4 5
s/d
Figme 24: Comparison of p~edicted su f cc p~essme
dish~butiom fornatucllycavitsti gaxi mmetric flow
over c conical cavitator with cylind iccl sfterbody. 5 =
03 Coarse(65xl7),medium(129x33) mdfin
257x65) mesh solu i ms are plotted
It should be noted 6~t du mg this
investigation, stesdy tate res its (time mtegations
based on ~t = = ) usmg UNCLE-M have been foumd to
be quite comistent with arithmeticclly averaged time-
dependent results This ~esult is expected to be u eful m
expeditmg the fuu e mtemretation of complex 6 ee-
dimensional flows in cddition, reel si gle phcse flows,
et the R y olds numbers considered, over these
axi mmetric bodies cc in fct umstecdy Howewx,
with the g ids md level of modeling cpplied he~e, 6he
UNCLE-M solutmns tended to be stecdy it seems
poss~ble that in recsed resolution md in omorction of
low R y olds number tmbulen e modelmg w mid
resolve 6his issu
Conclusions
The effect of Rey olds m mber on the ~esults
for 6he h misphericcl cavitator was not mticipated it
was cssumed that with the cppropricte mplication of
the high Rey olds number tmbul nce model et th wall,
the inviscid extenurl flow would dominate the flow-
field, determinmg cavity shape md si:D: (ie suLce
pressue) How ver, it cppears that shong flow-field
interacti ms du to the highly tmbulent separcted
closure ~egion are impo t mt to determming 6he
umstecdy mode To some extent, based on the avemge
re mlts, these phenomena are being ecu stely ccptmed
How ver, there are shortcommgs m 6he cunently
employed level of single-phcse tu bulen e modelmg
The validstion cases e smmed have
demon trsted the ccpabilities of UNCLE-M over c
rmge of importmt flow conditions The most
promin nt ~esult for validation is that th umstecdy
flequ n ies obtain d in mmmericcl remits cppear to be
boumded by 6he experimentcl data of Stmebrmg (1983)
for cll the mod led cases ther quclitative observations
mcd regarding 6he modeled case of the Occliber
cavitator et c cavitati m m mber of 0 3 mggest that
UNCLE-M hcs 6he ctility to ccrrectly represent 6he
overcll natme of umstecdy. complex, multiphcse flows
without n cessarily ccptu ing some of the fmer flow
detsils This m itseif is c validsti m of the cpproach
tak n h re Validsted mod Img based on parsmeters
reiated to profile d cg, cavity ienf h, cavity shape, md
t~ends of Stro~l flequ n y with cavitation m mber
hcs been cc mplished it seems clear 6~t wi6h hig)~er
fidelity tubulen e md mass t msfer modeling md
subsequ nt improved modeling of the closme ~egion, c
ben flt to 6he modelmg of umstesdy cavitating flows
wo id be obtcin d How ver, 6he cunent cpprocch has
cllowed rendering of unstecdy multiphcse flows et
R y olds m mbers relev mt to engin ering cpplications
in c m odeling method smenable to compl :x geomeh ies
md des~g cpplcatmns
The cuthors contmu to develop 6he
ccp~oilities of UNCLE-M This in ludes the pu suit
relev mt validation cases for compl :x th ee-d6m nsional
flows in cdditi m, n w levels of physiccl modelmg,
such es compressible phcses vie isodhermcl md full
en rgy modeling, will be in crporcted These n w
ccp~oilities, in cddition to 6he cl~ecdy in crporcted
ctilities to model buoyancy md ventilation, cc criticcl
12
to c cunent research goal, fhe ful co figmation
modeling of c high speed supercc itating vehicle
umdergomg mcneu rs
Acknowledgments
This work is mpported by fhe Office of Naval
R search, contmct kN00014-98-0143, wifh Dr Kam
Ng es contmct monitor
References
Brennan, CE., Ccvit~tion md Bubble Dvrumics,
O ford University Press, New York, 1995
Garabedfian, P.R., Colculation of Asiolly S mm tric
CoviLes and Jet, Pao J. of Mcfh 6, 1958
Grogger, H.A. & Alajbegovie, A., Colculahon of the
Covit tingFlowinVmturiGeom tnesUsingTw Fhid
Model, ASME Pcper FEDSM 98-5295 1998
Jameson, A., Schmidt, W., & E. Turkel, Numencol
SoluLons of theEuler Eg uoti ms byFinit VohmeMe6h
od Using Runge Kutto Tme St pping Sch mes, A AA
Pcper 81-1259, 1981
Kunz, Robert F. et al., Multi Phose CFD Anolysu of
Noturol and Vmtilot d Covitohon obout Subm ged
Bodies, ASMEPcperF DSM99-7364,1999~
Kunz, Robert F. et al., A Pnecondihoned Novier
Stokes Methodfor Tw Phose Flow wi 6h Applicoh on to
Covit tion Pr d icohon, A AA Pcper 99-3329, 1999 ([1)
to be pub lished in Comruters md Fluids
May, A., Wote r Ent y and the Covih Running Behov
iour of Missles, Na val See Systems Comm md
Hyd oballi tics Advisory Committee Techmiccl R port
75-2, 1975
Merkde, C.L., Feng, J., & Buelow, P. E.O., Computm
tionol Modelingof th Dyn mics of SheetCavitotim,
3rd h temational Symposium on Cc itati m, G en ble,
France, 1998
Orvag, S.A. et al., Rmo~lizohon GnoupModeling
and Turbulence Simulohons, Near Well Tu bulent
Flows, Elsevier Scien e Publishers B V, Amsterdam,
TheNetherkmds, 1993
Reichardt, H., The ws of Covit tion Bubbles ot Asi
ol ySymmet icol Bodies in oFlow, Mini try of Al cmft
Produ tion Vo kem ode, MAP-VG R ports md Tr msk -
tions 766, Of fice of Na~l R search, 1946
Rouse, H. & MeNown, J. S., Covitotion and Pnessu~e
Dut ibuLon, Heod Forms ot 7em Angle of Y w, Studies
in E gmeermg Bulletin 32, State University of lowc,
1 948
Sehdichting, H., Boumdarv-Lawv Theorv, M Gmw-
Hill, N w York, 1979
Stindh ring, D.R., Bdlet, M.L., & Holl, J.W., An Inrer
tigotion of Covi y Cycl ing for Vmtilot d ond Noturol
CoviLes, TM 83-13, The Pemmsyl mic State University
Applied Research Laboratory, 1983
Sthdhrhg, DR., Sccling of Ccvitation Dcmcg, M S
Thesis, The Pemmsylv mic Sbte Uniwvsity, University
Park, Pemm yl mi~, Augu t 1976
Taylor, L. K., Arahshahd, A., & WhdtfiAd, D. L.,
Unsteody Th ee Dimensionol IncompnessibleNoviem
Stokes Computohonsfor o Pnolote Sphenoid Unde going
Tm Dep md mtMoneurerr, A AA Pcper 95-0313,
1995
13
DISCUSSION
'' Shyy
University of Florida, USA
Inthis pap u She mthorshave summari edavast
amount of information resulting from Heir :-
search in developing Ed refming c CFD tool for
single- Ed multi-phcse flows The r suits have
been impressive in particular, it seems that mm-
less She fl O..- Is massively separated Ed or cavi-
tated, the present CFD tool c m perform quite
w 11, esp ciclly in temms of pressure coefficients
Ed main time-dependent features The mfhors
should be con hammed for their accomplish-
ments to date
With regard to the unresolved issues, Here are
se- oral that or c m Rome First, the mule phs e.
time dependent (for the ensemble averaged
qu mtities) turbulent flowsis obviously e major
challenge On the or h Ed, there me interactions
between different physical mechsni ms which
produce, dissipate, cormect, Ed diffuse the tur-
bulent kmetic en rgy Ed the Rey olds shess
compor nts, on the other h Ed, there is subst m-
tial mass, momentum, Ed en rgy exch mge be-
tw en liquid Ed vapor phases The resulting
physical framework is extr mely complicated
beyond what w have been Cole to pa diet with
adequate co flder e There is no quick, prscti al
solution to handle this challenge How ver, to
the least, models capable of hurdling (i) sub-
st mticl depart r from quilibrium betw en pa -
duction Ed dissipation of the turbulent kinetic
en rgy, (ii) misohopy between mom Rey olds
sin ss compor nts, Ed (iii) turbuler e-erJkmced
mass tr m fer across the phase interface, should
be emphasized
The second issue is related to the r ed for ~ -
solvmg the liquid-vapor boundary with due accu-
racy This issue is difhcult to h mdle bee mse the
interface's location, shape, Ed velocity must be
computed es part of the solution, resulting in c
syst m that doesn't have either c pa determir d
co figuration geometrically, or c fixed mass,
momentum Ed en rgy budget withm its domain
A accurate Ed robust interface tracking sch me
c m help improve She performance of the present
CFD tool
The third issue is related to the mmmericcl eb-
ments, in luding feet res such es dynamic cdap-
tation of the grid to help mcmtam den tble
resolution, satisfactory conhol of mmmericcl dis-
persion Ed dissipation in view of the highly
cormective multi-phcse flows, Ed way to e. p e-
dite Ed tabili e the computttior~tl procedures
Suffice it to say that She hors have Greedy
developed c highly impressive Ed effective
CFD capabilities in each of She issues discussed
clove, efforts are being made to help f rther
improve its performance in various difhcult Ed
import mt application areas To help develop
these advanced cmctilities, or mu t appreciate
the r ed for acquiring experimental i fommation
withadequcteresolution md compreh nor en ss
For example, t rbulent qu mtities, precise mter-
face deli nmn Ed con- ffnon-d~flunon ratios
are some key i fommation Nat to date, we have
not been Cole to document based on it tt-htnd
experimental i formation
Author's Reply:
Professor Shyy has made several valuable com-
ments regarding She difficulties of resolving
complex, multi-phcse flows With r gard to his
suggestions for improved turbuler e mod in ~
the mfhors suspect that She in orporction of such
models lies in the futur of fi is Ed other Rey-
nolds-Averaged Na vier-Stokes based efforts The
mthors pram to continue to incorporate improved
turbuler e mod leg in addition, the mthors
hope to in orporcte some better form of turbu-
ler e erJkmced mi ing Particularly of mterest to
the mthors, in the context of She cure nt model-
ing method, is the proper physics to apply m the
preser e of multiple gaseous spemes Ed c smgle
liquid species
The mfhors consider the second Ed third issues
raised by Professor Shyy to be r cessarily i
Icted it is believed Nat, for complex en m ering
co figurations, with current computstior~tl lim-
its, c r ssor~tble way to ccpt re sheet cavities is
by application of c method similar to what has
been applied her; i e the interfaces to be ccp-
tured will be c solution to the home en ous
mint re flow equations, possibly with multiple
species, mass h msfer, Ed equations of state
Thus, the inte face will be finite Ed sharps ss
will be grid dependent This is srLtlogous to the
mo t popular methods of shock capt ring for
en mee~mg purposes during modeling of com-
pressible flows As Professor Shyy has noted,
when such a method is combir d with grid td-
sptition, signiflcmt improvement in solution
quality may be achieved
DISCUSSION
J. R. dwa do
North Ca olina State University, USA
This paper details the validation of c sophisti-
cated CFD approach for modeling mcompressi-
ble, unsteady cavitating flows Attention is cor-
rectly focused on resolving the time-dependent
aspects of cavity fommation Ed growth, es such
processes are i herently unsteady Ed should be
modeled es such The approach is shown to pa -
diet time-averaged su face pressure distributions
togoodaccord Discrpcnciesevider edmaybe
c consequer e of the modeling but it mu t be
kept in mind chat the Rouse Ed McNown date-
base is over 50 years old, Ed measurement tech-
niques have improved substmticlly over the
years The model also predicts bubble shapes
that correlate well with moo r cent experimental
date, giving co fider e in its civility to resolve
the buk features of axisymmetric sheet cavity
Ho.. IN ids
The unsteady validation of the model is pr-
sented es plots of d cg coefhcient versus time,
with dominant frequer ies exhacted fiom the
signal by c spectral analysis While the corn et
to nd of m in r use in He Strouhal mmmber with
in r using cavitation mmmber is evider ed, the
actual values are not in accord with experimental
date The mfhors conjecture that These deviations
may result from se- eral factors, in luding insuf-
ficie t grid resolution particularly for higher
cavitation mmmbers) Ed He quality of She turbu-
ler~e model Factors that also could i Sued e
these comparisons in lude again the quality of
the experimental date Ed th ee-dimensior~l
effects There is certainly no guar mtee that the
cavity motion will rmcin axisymmehic over
time 2
Some questions that might be posed to the
mthors during She discussion section are as fol-
Iows:
I Are there my prams to rep at my of the ccl-
culations es th ee-dimensior~l runs? It would be
inter sting to see if the unsteady results ch mge
2 The ~ hors employ m empirical rate equation
to model the con- erimn of liquid to vapor Ed
vice verse es the pressure d ops below the vapor
pressure How sensitive are She results obtain d
to the rate coefficients, particularly es regards the
fir e-a-em ed predictions?
3 The ~ hors note Nat " tecdy-state" r suits
(using ve y large time steps) for surface pressure
di tributions are "quite consistent with arithmeti-
cclly averaged time-depffndent results" p 12)
Does this comment apply to the wake predictions
es w 11? It would appear that ccpt rmg the large-
sccle motion of the re-entr mt jet would be e;-
senticl in predicting he conect delayed r covery
of the pressure I would fhmk Nat the "steady"
calculations would predict c more cbmpt recov-
ery
In eon lusion, I find She mthors' work to be truly
representative of the i~a~e-ol:~h -art in cavitation
mod leg Only She extension to thee dimen-
sions Ed the validation thereof is required b:-
fore c high-fidelity tool for unsteady cavitation
prediction will emerge
Author's Reply:
Professor Edwards makes insightful commentary
regarding the th ee-dimensiorslity Ed tr msient
nature of sheet cavitation it is clear that he has
spent c greet deal of effort studying Ed model-
ing such flows Responses to his th ee questions
me listedbelow:
l
Subsequent to the comments of Professor
Edwards, She mthors have begun to under-
take some f ee-dimensior~l modeling of
the ogive cases that w re originally mecs-
ured experimentally by Rouse Ed McNow
Farticlly completed results me included here
in Figure A These results seem to indicate
the preser e of thee-dimensiorsl modes
How ver, c complete study, in luding sen
sitivity to smell mgles of attack, was not
reedy es of the time of the deadly for this
reply
The rate equation used for mass h msfer is c
w ok Imk m our model However, it hr.
been applied m c consistent maimer The
rate coefhcient was original y chosen em-
piricclly es or which produces cppro i-
mately correct stecdy-state cavity si e for c
given ogive et c specific cavitation mmmber
After this initial calibration, She rate coeffi-
cient hr. been left unch urged for all com-
putations This consistent application of the
rate coefhcient should allow results to be
fauly sssest d No sensitivity study has
been prefommed on d is value
3 The original eon lusions of the ~ hors :-
gardmg the unsteady calculations r fleet c com-
parison of the nme-c- em ed unsteady computed
pressur field to the steady state computed pres-
sure field on the surface of the ogi es Thus it
appears to the mthors that the computed m
steady motion of the reenh mt wake is captured
in c maimer that approximates he computed
steady- tate average Ed the average yielded by
the datccollectedbyMcy
DISCUSSION
El. Fa fell
ARL, The Per sylva la State Urlver-
slty, USA
First, I would I he to con r~mh~e my colleagues
for m out t mding modeling effo t in the area of
time-dependent multi-phase flows, which are of
particular mterest in the hyd odyrumics of m-
derwater high speed vehicles I cppl md Heir
persister e to push Heir analysis to the realm of
en m ering usefulr .., where He true utility of
the computttior~tl tool c mbe realized
The success of th simplified mass h m fer mo d-
els is notable Whet is the r~tur of the higher
fidelity mass tr msfer models that you are con-
sidermg Ed their intended t en fit? Please di -
cuss He relative merits of advanced turbuler e
modeling versus mass tr msfer mod in g in im-
proving the prediction of the unsteady cavity
dynamics
Author's Reply:
The mfhors me pleased to receive such fa vorable
commentary fiom Dr Farrell it was due to his
expertise in the field of cavitation in eption Ed
cavitation modeling that it was suggested that he
be m in ited discusser
Regrettably, the mthors have not been Cole to
advance She physical quality of mass h msfer
modeling beyond the simple rate coefficient
based model presented in the text it is hoped
that m in eption model may be developed with
physics based on the Jaundice of cavitation
mmclei in She flow This might be similar to work]
thathcsbeendor byDr Farrell[1]
The effect of turbuler e mod in g on the civility
to capture c cavity flows is suspected to be
strong Some of the short omings of the wall-
fur tion based, two~quction approach when
applied to single-phcse flows have been com-
mented on here by Professor Shyy Ed elsewhere
by Wilcox [2], Ed others A greet deal of :-
search ht. been devoted to turbulent flows Ed
turbulent modeling Thus, it is unlkely Butt. in
the r ar future, c signifc mt improvement in
applicable t rbuler e models will be developed
In comparison, She mass h msfer model em-
ployed is suppo ted by far less research Thus, it
is su pected that, in the r ar term, improvements
in applicable mass tr msfer mod leg will be
found Ed in orporcted into She mmmericcl
model it may then become clear whether im-
provement in quality of r suits is cttcir~ble by
improvement in mass tr m fer mod in g
DISCUSSION
R. Ar dt
Urlverslty of Mlr esota, USA
1) How do you explain the discrepancy
between frequer y of pulsing cavities d-
temmmed mmmericclly Ed experimentally?
2) Have you carried out my crurlysis of
partial cavitation on hyd of oils?
Author's Reply:
The mthors suspect Nat turbuler e mod in ~
mass transfer modeling, Ed th e-dimensiortl
effects all contributed to the k k of absolute
agreement with experimental unsteady r suits
Of these possible avenues of improvement, the
mthors have included some f e-dimensiortl
results m Figure A A preliminary examination
of these results do indicate the preser e of cddi-
tior~l modes However, et the decdlme for sub-
mission of f is reply, results are not yet com-
plete
The mfhors do also intend to apply the current
Ed fur re version of the computstior~tl model
on ocher en treeing co figurations including
partially cavitatmg hyd of oils However nor of
this modeling is yet complete
DISCUSSION
]. Cc lk
West Vlrglrla Urlverslty, USA
Strouhal mmmber, i e the frequer y of the pn-
mary vortex shedding, c m be easily pa dieted by
RANS codes if they are 2 order in time A
more appropriate validation is to compare the
centerline velocity variation in the value of a
bluff body.
The code treats volume fraction of gas as a pas-
sive scalar and solves mixture momentum equa-
tions with variable density. In this regard it is not
truly a multi-phase flow code. The slip between
the two phases were not accounted for, and this
can have significant consequences.
Author's Reply:
The authors thank Professor Celik for his n-
sightful comments. Unfortunately the centerline
velocity data for the experimental results used
for comparison was not available. The authors
readily acknowledge that the absence of a slip
model renders the method less capable of cap-
turing certain cavitation phenomena. This would
probably manifest itself in flows dominated by
bubbles. However, flows that the authors have
concentrated on here are essentially phase sepa-
rated, actually sheet cavities. For this type of
flow, it is hypothesized that the inability to prop-
erly represent certain bubble physics is insignifi-
cant. A wide range of cavitating flows may be
properly represented by homogeneous, equilib-
rium two-phase models. Successful researchers
have modeled unsteady sheet and even cloud
cavitation with homogeneous models: See, for
example, Arndt, Song, et al. t34.
In the current work, the volume fraction is not
thought to be a passive scalar. Here liquid vol-
ume fraction, id, is solved for in the liquid vol-
ume continuity equation as a dependent variable.
It appears in the momentum equations via the
formulation of Am and m Mass transfer from
liquid to vapor and from vapor to liquid takes
place due to source terms in the continuity rela-
tions. Thus ~ is fully coupled to and interde-
pendent with solution of the flow field. If, for
example, a different ~ field is created due to
alteration of some flow condition such as Rey-
nolds Number or cavitation number, the rest of
the solution flow field will be significantly i-
tered as well.
DISCUSSION
H. Kato, Tokyo University, Japan
Estimation of vaporization/condensation rate is
important when we analyze cavitating flow b=-
cause it decides the amount of vapor in the flow.
I'd like to know how the authors decided the
mass transfer rate at the interface between vapor
and water, and how the authors verified it.
Author's Reply:
The reply to Professor Kato's question has been
given in the earlier reply to Part 2 of the ques-
tions by Professor Edwards.
Figure A: Three-dimensional, wall-function
based, turbulent, unsteady, two-phase result.
1,245,184 cell grid. Modeled flow (from right to
left) over blunt ogive (shown in gray) with an
isosurface of volume fraction, ~=0.9, colored by
velocity magnitude on a field colored by velocity
magnitude. s=0.30. ReD= 1 .46x 105. Integration
time step size Unit /D = 0.00685
a)Ut/D=6.85
b) U. t/D=10.27
Figure appears to show the capture of an ~n-
steady and three-dimensional sheet-cavity flow.
References for Discussion:
I Farr 11, K J. An Euleriar~/Lcgnmgi m
ComPutationai Analvsis for fhe Prediction
of Ccvitation in eption, Ph. D Thesis, D -
putment of Mech miccl md Nuclear E gi-
n ermg, Pemmsyl mic State University,
August 2000
2 Wilcox, D C, Turbulen e Modeling for
CPD, DCW Indushies, Lc Canadc, Califor-
nr, USA, I 99S
3 A ndt, Song, et cl, Inst bilih of Portiol
Covit tion: A Nurnericol/Experirnentol Ap
prooch, ONR 23 Symposium on Naval
Hyd odynamics, val de Reuil, Pmnce, 17-22
Septermber, 2000
