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OCR for page 534
LES of Bubble Dynamics in Wake Flows
T. Celik, A. Smirnov, S.Shi
West Virginia University
Department of Mechanical arid Aerospace Engineering
Morgantown, WV26506-6106
Email: andrei @ smirnov.mae.wvu.edu
Nomenclature
DNS
LES
LPD
NS
RANS
RFG
SGS
~ Abstract
Direct Numerical Simulation
Large Eddy Simulation
Lagrangian Particle Dynamics
Navier Stokes
Reynolds Averaged NS equation
Random Flow Generation
Sub-Grid Scale
Length scale of turbulence
Time scale of turbulence
In this paper we present the results obtained using the
Lagrangian particle dynamics (LPD) method and a ran-
dom flow generation technique (RFG) developed by the
authors in conjunction with large eddy simulations (LES)
in application to turbulent bubbly wake flows. The hy-
brid LES+LPD+RFG approach was applied to the case of
a two-phase bubbly mixing layer and the high-Reynolds
number bubbly ship-wake flows. The simulations were
performed for the wake of a widely used ship Navy model
5415. We also present the results of a validation study
where the same method is applied to the bubbly mixing
layer experiments. The dynamics of the vortices and bub-
ble concentrations reproduced in simulations are in a good
agreement with experimental data.
2 Introduction
Lagrangian particle dynamics (LPD) method is practi-
cal and efficient in computing dispersed phases in multi-
phase flow systems. It is usually favored over Eulerian
two-fluid methods in the cases with dilute suspensions or
large concentration variability of the dispersed phase (El-
ghobashi, 1994; Crowe, 1998~. This is true for bubbly
wake flows, such as ship-wakes, where bubbles experi-
ence preferential concentration and clustering effects in
the near wake region and are rather dilute in the far wake.
The LED model is also conceptually simple and method-
ologically robust providing a numerically stable statisti-
cal method for evaluating dispersed phase statistics and
particle/bubble distribution functions. The algorithm for
particle tracking and population dynamics developed by
the authors demonstrated the ability to efficiently simu-
late large populations of particles including coalescence
effects with even modest computer resources (Shi et al.,
2000a; Smirnov et al., 2000; Smirnov and Celik, 2000~.
Simulations of bubbles in turbulent shear layers re-
quire accurate representation of the fluctuating flow-field
that governs bubble dynamics. When conventional RANS
(Reynolds Averaged Navier-Stokes) models are used this
accuracy is often lost or comes at a cost of empirical phys-
ical sub-models, e.g. turbulence. On the other hand,
direct numerical simulations (DNS) or LES require no
or a relatively simple subgrid-scale (SOS) model. LES
has the advantage of being able to handle flows with
higher Reynolds numbers and still accurately reproduce
large-scale turbulent structures (eddies), which are im-
portant in bubble dynamics. The majority of simula-
tions of turbulent bubbly shear layers using Eulerian-
OCR for page 535
Lagrangian approach is done using RANS methods for (Ad) and lift (Al) given by the following expressions
high Reynolds number flows (Joie et al., 1997; Mu-
rai, 2000) or DNS-type simulations for relatively low
Reynolds number flows (Elghobashi and Lasheras, 1996;
Ruetsch and Meiburg, 1994; Okawa et al., 2001; Murai
et al., 2001). An LES-based approach pursued in this
study provides the basis for multi-phase LES of large-
Reynolds number flows, like ship-wakes.
A major disadvantage of the Lagrangian approach is
the perception that an extremely large number of particle
trajectories is needed to obtain sufficiently smooth statis-
tics. This study elaborates that this is not necessary true as
long as the flow can be categorized as "nominally dilute".
3 Method
The LED algorithm for particle tracking and population
dynamics developed by the authors demonstrated the abil-
ity to efficiently simulate large populations of particles in-
cluding coalescence effects with even modest computer
resources (Shi et al., 2000b; Shi et al., 2000a; Smirnov
et al., 2000; Smirnov and Celik, 2000~. In this study the
LED algorithm was combined with the Large-Eddy sim-
ulation approach (Smirnov et al., 2001a; Piomelli, 1999)
to compute the distribution of bubbly phase in the near-
wake flow of a ship-model. The LES technique (Piomelli,
1999) was enhanced with The RFG procedure (Smirnov
et al., 2000; Shi et al., 2000b) was implemented into the
LES method to enable the appropriate representation of
initial/inlet turbulence conditions and as a subgrid scale
for particle dynamics solver.
The combined LES/RFG method was validated on the
case of turbulent mixing layer, and used later to simu-
late the wake of a Navy 5415 ship model as described in
(Smirnov et al., 2001a). The fluctuating velocities at the
inflow boundary provided by RFG were generated from
the data obtained in prior RANS calculations (Larreteguy,
1999~.
To compute bubble dynamics the following equations
of motion were adapted from Shridhar and Katz (1995)
~Ub =Aa+Ab+Ad + A~ (1)
A —
a—
3 (aatf + (Uf V)Uf)
Ab = -2g
Ad = 4r Cd | Urel | Ure
Al = 4 Cl | Urel I U ! (o ~
c') = 0.5*(VxUf)
where rb is bubble radius, Uf is fluid velocity, Urel is
the relative velocity of fluid and bubble, i.e. Uf—Ub,
and Cd ,Cl are drag and lift coefficients respectively. The
above relations were obtained for spherical bubbles and
by considering the limit of small air-to-water density ra-
tios Pb/PI ~ 0. The coefficients of drag (Cd) and lift (cl)
are themselves empirical functions of Urel and bubble di-
ameter, which were obtained from experimental measure-
ments of bubbles dynamics in turbulent vortices. The Bas-
sett term involving the history integral was neglected in
this case, following the conclusions of Shridhar and Katz
(1995) on small contribution of this term compared to the
buoyancy term.
Equation (1) was discretized using a second-order
Runge-Kutta time-stepping procedure. Bubble tracking
throughout the computational domain was simplified by
using a uniform Cartesian grid, so that the index of the cell
containing the bubble could easily be obtained using sim-
ple division by modulus operation. Fluid velocities were
interpolated to the location of a bubble using tri-linear in-
terpolation formula.
Bubble dissolution effects were accounted for by the
following equation (Hymen, 1994; Carrica et al., 1998)
dm 6~C —Co)k2/3 ~ u ~1/3 r4/3 (2)
where r is the bubble radius, ~ u ~ is the relative air-
bubble velocity and k is the diffusivity of air in water.
Constants CoO7Csurf represent the concentrations of dis-
solved gas at a distant point and at the bubble surface re-
spectively and were derived using Henry's law (Carrica
et al., 1998) for air solubility in water.
where the A vectors on the right-hand side represent ac- In a joint LES+LPD+RFG approach the flow and
celerations due to added mass (Aa), buoyancy (Ab), drag the particle solvers were using different time-stepping
2
OCR for page 536
schemes with independent selection of time-steps. Usu- 7c~is
ally sub-cycling of particle iterations was required to
reach a stable solution.
4 Results
4.l Validation on a turbulent mixing layer.
The validation of the LES/LPD approach was done on ex-
perimental data of a mixing layer (Rightley and Lasheras,
2000) (Fig.1~. The computational domain size is 0.55m x
0.2m x 0.2m in stream-wise, vertical, and span-wise di-
rections respectively. The mixing layer is generated by
two separate parallel flows with different incoming veloc-
ities. A thin flat plate, which is O.l5m length, 0.003m
height, and 0.2m width (the whole span-wise extent), is
mounted in the middle of the inlet plane. The velocity
of the lower half flow is 0.28m/s, while the upper half
value is 0.07m/s. Bubbles of 40,u diameter are carried in
from the lower half. As it was the case in the experiments,
a sinusoidal perturbation with the amplitude of five per-
cent of the mean flow, was added to the vertical velocity
component of the flow in the lower half of the inlet plane
during the simulations. The grid used in this simulation
was a 194 x 66 x 42 uniform grid on all three directions.
The cell size is 2.8mm x 3.0mm x 4.8mm. Free gradient
boundary condition was used for the outflow boundary.
Slip-wall boundary conditions were used at the top and
bottom (y-direction). In the span-wise direction (z), pe-
riodic boundary conditions were applied. At the surface
of the flat plate, slip-wall boundary conditions were used.
The reason for using slip-wall boundary conditions is that
in this computation the exact resolution of the boundary
layer was outside of the scope of the current study. More- The predicted rms velocity in axial direction is gener-
over the mixing process is dominated by the effect of the ally lower than the measured values. This leads to under-
shearlayer. estimation of turbulence intensities. Farther away from
the region where the mixing occurs, the difference is even
larger. The slip-wall boundary condition at the top and
bottom boundaries may be the primary reason for the ob-
served differences. Very little mixing occurs in these re-
gions, so the turbulence intensity is almost zero. A simi-
lar phenomenon is observed for the vertical mean veloci-
ties and their fluctuations (Fig. 2(b)~. It should be noted
that the rms values shown in Fig.2 represent only the re-
solved part of velocity field reproduced by LES, which
Figure 1: Mixing layer experimental setup.
locity, u' = t(2/3)k]~l2, which is used as an input to the
RFG method:
u' = (2/3) ~l2(cs/c~u~s/\2/(a~m) (3)
where Cs = 0.04, C,~' = 0.09 are the turbulence con-
stants, S is the strain rate tensor computed at each point,
/\ = 3 mm is the grid cell size, 6,'~ is the mixing layer thick-
ness, and a = 0.035 is an empirical constant.
Because the volume fraction of the bubbles was kept
small (~ 5 10-4), the effect of bubbles on the carrier flow
was not considered. This approximation is also supported
by Elghobashi and Truesdell (1993) where they indicated
that the influence of such low dispersed phase particles
(bubbles) would be seen in the smallest scales, not in the
large, energy-containing scales dominating the mean and
rms velocity profiles.
Figure 2(a) shows the stream-wise mean velocities and
the fluctuations at stream-wise position x/X = 1.25, where
~ = 79.36mm is the length of Kelvin-Helmholtz instabil-
ity wave, representing the characteristic flow length scale.
The profile of the mean stream-wise velocity is in a good
agreement with the measurements.
Central differencing (CD) scheme was used to dis-
cretize the convection term. The the Smagorinsky SGS
model was used to represent the effects of subgrid
scale motion on the carrier flow and the RFG technique
(Smirnov et al., 2001 b) was applied to reproduce the
effect of turbulence on the bubbles. In the latter case
the standard relation for turbulent eddy viscosity, v, =
C,Ukl/2 * Im''r' and the Smagorinsky model v, = CsA2S
were used to provide the estimate of the fluctuating ve-
3
OCR for page 537
OCR for page 538
domain with time interval of 27 milliseconds for the du- the boundary layer. Some irregularity of computed data
ration of the 12.7 seconds, resulting in 47,000 bubbles that further downstream is due to statistical uncertainty of the
entered the domain during the course of the run. Fig- sample as bubble concentration becomes more dilute.
ure 5 shows a typical bubble distribution computed by It should be noted that, being statistical in nature, the
LES/LPD algorithm compared with those observed in ex- LED method enables the refinement of the histogram and
periments. The bubble cloud can be seen to be entrained improvement of the accuracy by subsequent accumulation
by the fluid entering the mixing region from the high of bubble statistics. Since the statistical error is propor-
speed side into the cores of the coherent vertical flow tional to 1/ni/2, where n is the number of bubbles in a
structures present in the mixing region. The pictures from histogram box, it would require four times as many bub-
the simulations represent an instantaneous distribution, bles for a two-dimensional histogram to double the reso-
which is different for any given time, but shows common lution along each axis, or to reduce the statistical error by
statistics and similar dynamical features as seen in the ex- half with the same resolution. The choice of histogram
periments. size is a trade-off between the spatial resolution and sta-
Spatial histograms of bubble distributions were ob- tistical error.
rained by counting all bubbles passing the cells of the
3D-histogram and accumulating the statistics. Normaliz-
ing the histogram data by the total number of bubbles in-
jected, gave bubble probability density functions and con-
centrations. The statistical error, a, in the number of bub-
ble counts, n, for each slot of the histogram can be esti-
mated from binomial distribution as c, = En(~1—n/N)] ~/2,
where N is the maximum number of bubbles injected dur-
ing the simulation. This will add to the uncertainty in the
layer thickness calculations for larger X, and smaller bub-
ble concentrations.
Figured shows the surface of the histogram and the
contour plot of the constant concentration levels. Fig-
ure 7 shows the growth of the mixing layer thickness in
the stream-wise direction (X) with and without the RFG
model. Over-prediction of layer thickness at small X is
due to the finite resolution of the histogram. The inclusion
of the RFG model improves the predictions at greater X.
Figure7 also shows predicted and measured develop-
ment of the mixed layer thickness. The overall agreement
is good except at large axial distances where a smaller
rate of growth of the mixing layer in the stream-wise di-
rection is observed when no subgrid scale model was ap-
plied. Although the inclusion of the RFG model (Smirnov
et al., 2001 b) improved the predictions of the boundary
layer growth at higher axial asymptotics, it also increased
the uncertainty in the boundary layer thickness because
of the higher dispersion in bubble distribution. Increasing
the bubble sample will improve the accuracy.
Figure8 provides the comparison of predicted bub-
ble concentrations against experimental data. The agree-
ment is reasonably good especially at the beginning of
s
4.2 Ship-wake simulations.
The ship-wake simulations were performed for the wake
of a widely used ship Navy model 5415 ~ (Smirnov et al.,
2001a; Carrica et al., 1998~. In the simulations a com-
putational 164 x 98 x 92 grid was used to represent the
near-wake region of one ship-length L in the axial direc-
tion, 0.3L in depth and 0.6L in the span-wise direction.
The RFG procedure was used to initialize turbulent fluctu-
ations in the inlet plane (Smirnov et al., 2001b). The free
surface was assumed to be flat, invoking the low Froude
number approximation. Bubbles with a uniform size of
lOO,u were randomly injected at the inlet plane. The prob-
ability distribution of the injection points was set propor-
tional to the turbulent kinetic energy level obtained from
the preliminary RANS solution. At the free surface the
life-time of the bubbles was set to zero for the absence
of more exact empirical data. The integration of Eq.~1),
via a second-order Runge-Kutta scheme provides the new
velocity, vitt), in the xi location for each particle as a func-
tion of time.
Figure 9 shows an instantaneous picture of the bubbles
in the wake as observed from the rear-bottom corner of the
computational domain (ship stern is not shown). It can be
seen that the bubbles experience a trend to a preferential
concentration regions. These regions are dictated by the
dynamics of the flow in the near wake. Figure 10 provides
a cross-sectional view of a instantaneous velocity distribu-
tions in the wake, which is rather typical for ship wakes
~ htip://wwwSO.~.navy.mil/54 ~ S
OCR for page 539
(Hymen, 2000~. Two symmetrical vertical structures in
the velocity distribution coincide with the bubble cluster-
ing regions. With the higher vortex intensities, which will
take place at a higher Reynolds number, bubble distribu-
tion in the wake will be increasingly affected by these vor- 5
tices. This is especially true for smaller bubbles.
Figurell shows typicalcross-sectionaldistributionsof The results of this study show the viability of joint
bubbles in the wake. These distributions are not instanta- LES/LPD method for computing turbulent bubbly wakes,
neous, but represent total counts of bubbles accumulated which can be applied to high Reynolds-number ship-wake
over the simulation run. flows. Realistic bubble distributions for mixing layers and
wakes were obtained.
The effect of bubble influence on the flow field was
not considered in this study. Nevertheless, the work by
others (Elghobashi and Truesdell, 1993; Truesdell and
Elghobashi, 1994; Elghobashi and Lasheras, 1996) indi-
cates that when there is a large density ratio between the
phases, such as that in this study, the influence of bubbles
on the carrier phase may become important even though
the void fraction remains small. Work on this topic are
continuing under the current project.
To further improve the accuracy of predictions for the
flat-plate wake a more realistic no-slip wall boundary con-
dition may be used together with the grid refinement at the
wall so as to resolve the turbulent boundary layer. More-
over a more realistic model of bubble/turbulence interac-
tion near the free-surface needs to be implemented in the
future.
for smaller bubbles (~ 50,u) and at lower depths (~ 10m).
This was also confirmed by LES simulations.
Conclusions and future work
PLANE COUNT RATIO
1 46998 0.01
2 9803 0.21
3 3326 0.07
4 1119 0.02
Table 1: Cumulative bubble distributions in different
planes
Table 1 shows the total bubble counts in different planes
Ci=l,4 and the corresponding normalized ratios: Ri =
Cinch. These distributions indicate a strong depletion
of bubbles over the half ship-length distance, shown in
Fig. 13. The depletion of bubbles is due to the buoyancy
driven migration to the surface and the dissolution effects.
Buoyancy forces affect mainly large bubbles, whereas the
dissolution affects primarily small ones. Therefore, even
the small bubbles entrained by the vortices will eventually References
disappear from the domain.
Figures 12 show the contours of probability density
functions of bubble occurrence in the wake. These
probabilities were computed from bubble-distribution his-
tograms obtained in the way as for the mixing layer vali-
dation case described above. The profiles shown in Fig.12
indicate a rapid bubble population decay and gradual
spreading of the bubble cloud on the distance of one ship-
length. Although the classified nature of bubble measure-
ment data in the wakes of Navy ships prevents us from
making a direct comparison, we believe that the predicted
bubble distributions are similar to those observed in typi-
cal ship wakes (Hymen, 1998; Hyman, 20001.
Separate estimates of bubble dissolution rate have
Carrica, P., Bonetto, F., Drew, D., and Lathey, J.: 1998,
The interaction of background ocean air bubbles with
a surface ship, Int. J. Numer. Meth. Fluids 28, 571
Crowe, C.: 1998, An assessment of multiphase flow
models for industrial applications, in Proceed-
ing of FEDSM'98, Vol. FEDSM-5093, Washing-
tow,DC,USA
Elghobashi, S.: 1994, On predicting particle-laden turbu-
lent flows, Applied Scientific Research 52, 309
Elghobashi, S. and Lasheras, J.: 1996, Effects of Grav-
ity on Sheared Turbulence Laden with Bubbles or
Droplets, in 3-rd Microgravity Fluid Physics Confer-
ence, Cleveland, OH
shown that its effect can be neglected in the case of mix- Elghobashi, S. and Truesdell, G.: 1993, On the two-
ing layer and can be noticeable in the case of ship-wake way interaction between homogeneous turbulence and
6
OCR for page 540
dispersed solid particles. i: Turbulence modification,
Physics of Fluids A 5, 1790
Hyman, M.: 1994, Modeling Ship Microbubble Wakes,
Technical Report CSS/TR-94/39, Naval Surface War-
fare Center. Dahlgren Division.
Hyman, M.: 1998, Computation of ship wake flows with
free-surface/turbulence interaction, in 22nd Sympo-
sium on Naval Hydrodynamics, pp 11-32, Washing-
ton,D.C.
Hyman, M.: 2000, Bubble Concentrations Near a Ship
Surface, Private communication
Vat de Reuil, France
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F1
Smirnov, A., Shi, S., and Celik, I.: 2000, Random Flow
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1 Y' . 1 ~ ~~ ~
Numerical study of the three-
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7
OCR for page 541
At:
Ad
:=
:"
- ~
O . 03~
O . 0 1-
_ O~
_.:
i.
_~.
A:
· ~
i_
~ C~
t: ~ ~ ~ ~ ~ : ' -] ~~ -I q ~~ :~- ~ ~ ~' ~ :~: by, ~ ~
~ ~ `~ ~ I: ~ ~ ~ ~' :` : ,, :~ ~~::J~ ::~;:~: 7~ ~ ~ ~ :` Or .' ~ ~ ~ ~
~.~i hi ~ ~ ~ ~ ~ ~ by: Aft: t. ~ :§ ~ ~ ~ ~ ~ ~
i:
..
.,
: _- ~ g:
: ~
: I:
0.04~ .
A: ~ ~ ~ id. A.: :~ : L.,
~~~~
i.
- =~C~
: :: : : :::: :-: : as, :: :: :: ::::: ~ ::: ::::: : : ~ : : ad.
: :: ::: ::~::: :: A::.. :::: : : ~ : : :: ::~:
' ~
. :: : I::: : :: ~ : ::::: : : : : :
. : :::
,—:— . ~
-
o': : ~ ~: -~ ~ ~: ~ ~, ~ :~ ~ ~ ~ ~ ~:~ :~: ~ '~ : :~:
'' ~ Ph~:~
~ t r
:~ ~ ': ~ :^ ~~: : :~ :
— ~ a. ~ ~ :Ir ~ : ~> ~L. ' AL
1
(a) Measurements by Rightley (1995)
. . . . . . ~ . . _ _ ~ ~ ~ ~ ~ t t t
- ' t ~ ~ ~ ~ ~ j ~ 6, . ~ t ~ ~ ~ ~ t ~ ~ ~ I ~ ~ ~ ~ ~ ~ ~ ~ t, ~ t ~ t t t t
O_ ~ ~ 27 1 0~129 1 0~31 1 0~33 1 0~35 1 0~37 1 0~39 1 0~41 1 0~43
0 180 360 540 720
x (m) / Phase Angle (degree)
(b) Simulations
Figure 3: Velocity vectors at x/\ = 2.50
8
OCR for page 542
0.14
0.05]
~ at
—0 . 05-
~9
Figure 4: Computed vertical velocity contours.
(a) Computed
(b) Rightley and Laseras (2000)
Figure 5: Instantaneous bubble distribution
9
Level v
23 8.5 E-02
21 6.3 E-02
1 9 4.0E-02
1 7 3.0E-02
1 5 2.4E-02
1 3 -2.6E-02
1 1 -3.5E-02
9 -5.0E-02
7 -6.1 E-02
5 -8.3E-02
3 -1.1 E-01
1 -1 .3E-01
OCR for page 543
'PDF'
1 .2
0.8
0.6
0.4
0.2
o
4
X/L
~/~//A///~////
3
0\
-1 ~
0.8
~0.6
j/~/~/
60 40 \/e°rtical distance [mm] -60
Figure 6: Normalized histogram of probability density function of bubble distributions.
60
50
40
cot
c,, 30
._
cat
.= 20
>
10
o
...
^-'3X<;'l"'"~""'
0 0.5
aid...+
Y.` - ~ at,
..~.., .
I;;; ;!
~,..~. -
,,,,,...,~,,~ 1
~ 1__;~
..,d'.~,
1 1.5 ~
Streamwise distance [X/L]
I. .---
,.H ~ ~ ~
...............
=.,,)~..~.~
_ 2.5 3
Figure 7: Mixing layer thickness.
3 measured; * - computed without RFG; x - computed with RFG
10
OCR for page 544
0.8
0.6
0.4
0.2
o
1.2 , , . . , 1 . 1.2
.--\\ Doer
, . . . , \ · · O
-40 -30 -20 -10 0 10 20 30
Vertical distance [mm]
(a) x/=0.31
40 -40 -30 -20 -10 0 10 20 30 40
Vertical distance [mm]
(b) x/0.63
0.8
0.6
0.4
0.2
o
A . ~ A \
~ .
''. . ~
'I.;, ~
.A ~
-40 -30 -20 -10 0 10 20 30
Vertical distance [mm]
(c) x/=1.25
1.2
0.8
0.6
0.4
0.2
o
1 ~
0.8
0.6
0.4
0.2
n
, , . , , , . 1 .1
1
1 At' . 0.9
\~ 0.8
~ \ it/\ o76
>- 0.3
, , , , , , me,, of
o
- 0 -30 -20 -1 0 0 1 0 20 30 40
Vertical distance [mm]
(e) x/=1 .88
Figure 8: Bubble concentrations (~0.079365 m).
-+- experiments, computations.
11
40 -40 -30 -20 -1 0 0 1 0 20 30 40
Vertical distance [mm]
. . . .
· ~ ~
~ \
Hi\
A\
· iK
(d) /1.56
.. ~
\ ;'
-40 -30 -20 -1 0 0 1 0 20 30 40
Vertical distance [mm]
(f) x/2.50
OCR for page 545
!:
.j
I..
Figure 9: Bubbles in a ship-wake. Rear bottom view.
-o.o,~
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
~ } ~ I, I, ~ ~ ~ ~ ~ ~ ~
, , , , , , ~ , ~ , .
, , ~ . . ., it,, ~
-0.05 0 0.05
Figure 10: Velocity vectors at X/L = 0.2
12
OCR for page 546
-0.01
-n nob
-0.01
-0.02
-0.025
-0.03
-0.04
-0.045
-0.05
-0.1 -0.05 n
(a) X/L = 0.01
~ I-:'
-0.015
-n.025
-n no
-0.045
-0.05 +
n ns 0.1 -0.1
item ~
+ ++++ + +++ *I+
+*+ +$+ + be+ ++
ti , 1 11$
-0.1 -0.05 0 0.05
(c) X/L = 0.50
t3
.,.~. ~ ~
-0.05 0 0.05 0.1
(b) X/L = 0.25
-0.015
-0.02 -t
-0.025 .+
-0.03
-0.035 .
-0.04
-0.045
;+~
At + ++ +` +++ ++{ ++~++
l++ Be++ + $+~++~$+ +
+ ++++ + + ++++~+
+ ++ +++ + ++
+
+t~
+$+ t
++ + + $ +
+
+
, , f+
0.1 -0.1 -0.05 0
(d) X/L = 0.75
Figure 11: Bubble distributions at different cross-sections
13
0-05 0.1
OCR for page 547
O, ,.-~6;' ,
-0,005 my,,!
0.01
-0.015
0.02
0.025
-0.03
0.035
-0.04
0.045
-0.05
-0.1 0.05 0 0.05 0.1
0.02 ---------
0.015
0.01 ~—
ohs . ~ . .
(a) X/L = 0.01
O :~.-': ·00005
ooo2o53 hi;',.
oooo4
0.045 .
0.05 1 , ,
-0.1 0.05 0 0.05 0.1
0.002 -------
).0015
(b) X/L = 0.25
0.0008 ~ 1 ~ 0.00025
0.0007 0.0002
0005 0005 ooos 0015-—-
55:, 0002 0OOO5~ :
-O.M ~ '' <~ jOOO1 -- ·0 02 ~ ~ ~ 1
ooo45 ~ 1 .l ~ 1 004 t 'I-- ~
0.05 -0.05
0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1
(c) X/L = 0.50
eO5 ~
(d) X/L = 0.75
Figure 12: Bubble probability density functions at different cross-sections
14
OCR for page 548
oc9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1 1~' 1 1 1 1 1 1
-
-
-
-
-
-
0.1
o
-
1 , , , , _ ~
0 0.1 0.2 0.3 0.4
X/L
Figure 13: Bubble depletion in the wake
0.5 0.6 0.7 0.8
Figure shows the ratio of total number of bubbles in each plane to the
number of bubbles in the first plane.
15
OCR for page 549
DISCUSSION
P. Atsavapranee
Naval Surface Warfare Center, Carderock, USA
Do you have any physical experimental data to
compare to your result of bubble distribution
around 5415 hull?
AUTHORS' REPLY
The question reflects a legitimate concern that
without proper validation the confidence level in
the simulations will low. Unfortunately there is
very little data available in the open literature on
bubble distribution in ship wakes. The data
known to us is not appropriate for comparison
with the present simulations. Other sources of
data seem to exist (Hymen, 2002) but they are
classified. There is, however, an ongoing
program under ONR to quantify bubble
distribution around the ship model 5415, but not
directly in he wake. It would be highly desirable
to conduct some carefully planned experiments
for measuring the bubble distribution in the wake
of surface ships or ship models to supplement
our computational work.
References
1. Hyman, M. (2002) Private Communication.
DISCUSSION
R.E. Melnik
Mississippi State University, USA
The authors should be commended for their
development of an effective approach for the
numerical solution of the very difficult problem
of predicting the bubble distribution in a
turbulent wake, a problem of great current
interest in the US Navy. The authors approach
based on coupling their LED and RFG methods
with an established numerical method for the
numerical solution of the 3D unsteady Navier-
Stokes equations with LES modeling, for dilute
bubble concentrations, is potentially the most
accurate approach actually implemented to-date.
The authors provided some validation of their
method through comparison with experimental
data for a flat plate generated wake that showed
good agreement for mean flow stream-wise
velocity but otherwise relatively poor agreement
for the rms velocity fluctuations. Although, these
results provide encouragement in the overall
accuracy of the method, much uncertainty
remains on the question its quantitative accuracy.
My discussion will focus on the uncertainty in
the accuracy of the method. Inaccuracies are
basically introduced from three sources:
approximations employed in the physical models
(LPD, RFG, LES), errors from the slip flow
boundary condition, and numerical errors arising
from the computational methods employed for
the LED and LES solutions. Because these errors
are highly intertwined it is really not possible to
draw meaningful conclusions about the accuracy
of the individual sources of error. This point was
forcefully stated in the conclusions of the 1981
Stanford workshop on the validation of
turbulence models in vogue in that timeframe. In
that workshop, numerical solutions of the
turbulent boundary layer and RANS equations,
using many of the available turbulence models,
were compared with experimental data for a
wide variety of flow conditions and geometry.
Since that time computing power has increased
to the point that it has now become customary to
check the accuracy of numerical solutions of the
RANS equations through grid refinement
studies. Indeed, some journals will not accept
papers on numerical simulation unless accuracy
assessment data are included in the paper. Today
it is equally important to assess the level of
numerical error before drawing conclusions on
the adequacy of LES models.
The authors of the present paper point to the
need to add a no-slip condition and grid
refinement to better resolve the turbulent
boundary layer emanating from the flat plate.
They also pointed to the need for a better model
of free surface effects on bubble turbulence
interaction. I think it is equally important to
conduct grid refinement studies on the numerical
solution of the LES equations, before drawing
conclusions on the adequacy of the physical
models employed in the computations. LES grid
refinement studies are more difficult than those
for the RANS equations, because of the need for
much greater computer resources as well as the
need to consider the two length scales that
appear in numerical solutions involving LES
models, namely the filter width, /\, employed in
the LES model and the mesh spacing, h, used in
the computational grid. This issue has been
studied in great detail in Refs 1-4, listed below.
These papers considered numerical solutions of
various LES modeled equations, including the
Smagorinsky model, for the relatively simple
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problem of a temporal mixing layer. The
Reynolds numbers employed in these
computations was low enough to permit
computation of highly resolved DNS solutions,
which then served as a basis for assessing the
accuracy of the LES solutions.
Here, I briefly summarize the methodology and
the main conclusions reached in those papers.
The quality of the solution depends on the
choices of the parameters, ~ and h. The papers
considered various values of these parameters
organized in terms of the ratio, r = i\ /h. If ~ is
held fixed while h is reduced a grid resolved
solution of the LES equations will be approached
in the limit ho 0. An alternative approach is to
consider the ratio, r, fixed, while decreasing h.
This would yield a grid independent DNS
solution as h approaches zero, although at
considerably higher computational costs. The
papers considered the convergence of both mean
and fluctuating quantities and they also provided
an analysis of the computational cost of the
various choices of ~ and h. The principle
conclusion of these studies is that the ratio, A/h,
should greater than 2 ~ r >2) and for some other
quantities be as large as 4 -5 in order to achieve
grid converged solutions of LES equations. Most
LES solutions in the literature, including those in
the paper under discussion, employ a value of 1`
/h = 1, which implies that these results are likely
contaminated by numerical error.
I realize that these conclusions, which are based
on relatively low Reynolds number
computations, and are for a different type of
mixing layer and therefore, may not apply to the
computations of the paper under discussion.
Nevertheless the results clearly establish the
need to demonstrate that numerical solutions
with LES models are free of effects of numerical
contamination.
I will close my discussion of the paper with a
question for the authors. The RFG procedure is
employed to provide turbulence inlet and initial
conditions as well as to provide a subgrid scale
model in the particle dynamics solver. The
subgrid scale model is used to provide a
turbulent flow representation to account for the
small scale dispersion of particles. It is possible
to couple the RFG technique to a RANS solver
over the entire flow domain and to thereby avoid
the need to use an expensive LES solver, greatly
reduce required computer resources. It would be
very interesting to compare the accuracy and cost
of such a procedure with that of the LES method
used in the paper. Have the authors considered
such an approach?
References
1. Vreman B., Geurts B., & Kuerten, H., 1996
"COMPARISONS OF NUMERICAL
SCHEMES IN LARGE-EDDY SIMULATION
OF THE TEMPORAL MIXING LAYER" Intl J.
Numer Meth. Fluids 22, pp. 297-31 1
2. Verman B., Geurts, B., and Kuerten H., 1997
" Large Eddy simulation of the turbulent mixing
layer", J. Fluid Mech. 339, pp. 357-390
3. Geurts B. J. and Frolich, J., 2001 "Numerical
Effects contaminating LES; a mixed story", in
Modern Simulation Strategies for Turbulent
Flow, edited by B. J. Geurts (Edwards, Ann
Arbor, MI ), pp. 309-327
4. Geurts B., Frohlich J., 2002 " A framework
for predicting accuracy limitations in large-eddy
simulation", Physics of Fluids, 14, pp. L41-L44
AUTHORS' REPLY
We appreciate the comments of this reviewer
with regards to quantification of numerical
uncertainty in LES (Large Eddy Simulation).
The reviewer seem to appreciate that this is not a
trivial issue. In LES calculations both the
modeling errors (i.e. the inaccuracies resulting
from the sub-gri-scale (sgs) model) and
numerical (primarily the discretization) errors are
directly proportional to the grid size. This is
especially true in cases of implicit top-hat filters
inherent to finite volume formulations such as
that used in the present paper. As the grid
resolution is refined a good LES should
approach to direct numerical simulations (DNS).
If very fine grid resolution is used in LES
calculations such that the simulation results are
close to those obtained from DNS it will not be
economical. If on the other hand relatively
coarse grids are used to take advantage of the
LES technique, then the discretization errors are
large and they pollute the sgs contribution.
Hence the user is faced with a dilemma of what
to use for an optimum accuracy and benefit from
LES. Although the present authors are aware of
this non-trivial problem, no attempt was made to
determine the numerical uncertainty in the
present LES calculations. However, work is
underway by our group to develop a practical
OCR for page 551
methodology with which the quality of an LES
can be measured. Preliminary results from this
effort will be reported in an upcoming ASME
Fluids Engineering Conference (see Celik et al,
2003).
We thank the reviewer for making us aware of
the relevant study by Geurts and Froehlich
(2002~. We studied this work closely and found
out that the method presented in that paper is not
directly applicable to our case. Since no DNS
results are available for our case, the calculation
of the "exact" solution is not possible. Moreover,
the activity parameter "s" suggested by Geurts
and Froelich requires the calculation of volume
averaged turbulent dissipation rate which
inherently includes both the modeled dissipation
and the numerical dissipation; segregation of he
two is necessary but not easy. As mentioned
above, this is the topic of a future work by our
group. In this approach we attempt to estimate
the percentage of resolved turbulent kinetic
energy relative to the total; if more than 70% is
captured we judge the LES as adequate.
Preliminary calculations have revealed that our
current LES results presented in this proceedings
satisfy this criteria. Simulations with much finer
grid resolution using parallel version of the same
code will be presented in the future Symposia on
Naval Hydrodynamics.
As for the last question, no we did not consider
running RAN S supplemented by RFG. There is a
good reason for that. The reason for using LES,
in the first place, was that RANS smeared out
even the relatively large coherent vorticies which
are important for bubble dynamics. RGF on the
other hand deals essentially with fine scale
turbulence. Hence, it would be counter
productive to resort back to RANS.
References
1. Celik, I.B., Cehreli, Z., Yavuz, I. (2003) "Index of
quality for large eddy simulations." Proceedings of
the 2003 ASME Fluids Engineering Division
Summer Meeting, Honolulu, Hawaii, July 6-10.
2. Guests, B.J. and Froehlich, J. (2002) "A
Framework for predicting accuracy limitations in
large-eddy simulation," Physics of Fluids, Vol.
14, NO. 6, pp.L42-L44.
Representative terms from entire chapter:
bubble dynamics
