-
Sci. Tech.• 1996. Vols. 113-114, pp. 329-350Reprints available
directly from the publisherPhotocopying permitted by license
only
({:) 1996OPA (Overseas Publishers Association)Amsterdam B.V.
Published in The Netherlands underlicense by Gordon and Breach
Science Publishers SA
Printed in Malaysia
Laboratory Simulation of Flamelet and DistributedModels for
Premixed Turbulent CombustionUsing Aqueous Autocatalytic
Reactions
S. S. SHY* Department of Mechanical Engineering,National Central
University, Chung-Li 32054, Taiwan, R.O.C.
R.H. JANG and P. D. RONNEY Department of Mechanical
Engineering,University of Southern California, Los Angeles, CA
90089, USA.
(Received March 1. 1996)
Abstract-Flamelet models have been widely applied to predict
premixed turbulent combustion becauseof their simplicity for the
description of chemical features in a turbulent flow field. We had
used anaqueous autocatalytic reaction which produced an
irreversibly propagating front with characteristicsclosely matching
many ofthose assumed by flamelet models, to simulate premixed
turbulent combustion.We then studied experimentally the influence
of turbulence on this reaction-diffusion propagating front.The
turbulence was generated by a pair of vertically vibrating grids in
a chemical tank and was found tobe nearly stationary and isotropic
in the core region between the two grids, as verified by laser
Dopplervelocimetry. Visualization of these turbulent propagating
fronts in the nearly isotropic region was ob-tained using the
chemically reacting, laser-induced fluorescence (L1F) technique. In
this paper, theseplanar LIF images were then processed to extract
the mean reaction progress variable (C), the variance,and the
probability density function (pdf) of the progress variable.
Markstein numbers of these chemicalfronts are probably close to
unity. Results of the progress variable pdf revealed a bimodal
distribution ata low turbulent Karlovitz number (Ka) with nearly
zero probability of intermediate values of c. At higherKa, the pdfs
seemed to show significant probability of partially reacted fluid,
in support of the theoreticaldescription proposed by Pope and Anand
(1984). Values of the turbulent burning velocities (ST) were
alsoextracted from successive images. When the turbulent Karlovitz
number is less than 5, measurements offront propagation rates (U T
ss STISL) as a function of the normalized turbulent intensity (U ==
u'ISL) showa roughly linear increase of UT with U. Values of U T
are found to be much lower than those proposed byBray (1990) and by
Pope and Anand (1984), but in good agreement with a Huygens
propagation modelemploying renormalization group analysis. At Ka
> 10, U T departs from the linearity and behaviour orthese
propagating fronts possibly suggests that modes analogous to
distributed combustion are observed.For high Ka, values of U T are
compared to a classic model. These results are also compared to
premixedgaseous experiments.
Key words: Premixed turbulent combustion, flamelet and
distributed models, aqueous autocatalyticreactions.
I. INTRODUCTION
The study of premixed turbulent combustion is of great practical
importance be-cause of its occurrence in spark ignition engines
(Heywood 1988; Bracco 1990) andits potential for reduced NOx
emissions in some gas turbine applications (Correa1990).
Furthermore, studies of premixed turbulent flames may be central to
theunderstanding of yet more complicated combustion phenomena
(Bradley 1993).It has been recognized for some time that different
modes of combustion, such as
flamelet and distributed combustion, may exist in premixed
turbulent flames (e.g.
·Corresponding author.
329
-
330 S. S. SHY et al.
Williams 1985; Peter 1986), probably depending on two
dimensionless parameters,for example a turbulent Reynolds number
(ReT) and a turbulent Karlovitz number(Ka) if Markstein number (Ma)
were unity. An adaptation of Eq. (23) from Bradley(1993) gives:
.
Ka =0.157 ;v ReTScu'l,
ReT = - ;vv
Sc=-D (1)
where U is the ratio of the r.m.s. turbulence intensity to the
laminar burning veloc-ity, Sc is the Schmidt number, I, is the
integral length scale of turbulence, and v andD are the
representative momentum and reactant mass diffusivities. Flamelet
com-bustion occurs at values of Ka« I while distributed combustion
may occur atvalues of Ka » I. In the former, the chemical reaction
time is much shorter than thecharacteristic turbulent time scale,
so that there are only two types of fluid, reactantand product,
separated by a thin flame front. Some models of premixed
turbulentcombustion employing this flamelet concept (e.g. Pope
& Anand 1984; Yak hot 1988;Bray 1990; Peters 1992; Kerstein
& Ashurst 1994) predict the effect of laminarburning velocity
of the planar steady flame front (SL) and turbulence
characteristicssuch as the root-mean-square (rms) velocity
fluctuation of the turbulent flow (u') onthe mean propagation rate
(ST) of the wrinkled (and possibly disrupted) flame front.Note that
model predictions vary considerably with no consensus on any
pointexcept that ST/Sl, increases as U'/Sl, increases, indicating a
need for benchmarkexperimental or computational data for
comparison. However, experiments (e.g.Abdel-Gayed et al., 1987) do
not agree closely with models. This is probably becauseexperiments
have been conducted in premixed combustible gases that do not
satisfysome of the simplifying assumptions the models usually
require for tractability.These assumptions may include: (I) the
effect of thermal expansion can be neglected(negligible density
change across the flame front), (11) heat loss is absent, (1lI)
ther-modynamic and transport properties are constant, and (IV) the
turbulent flow fieldis homogenous, isotropic, and statistically
stationary. Furthermore, direct numericalsimulations have only been
performed for small u'jSL (Rutland et al., 1990; Haworth&
Poinsot 1992) due to numerical accuracy limitations. Recently, Shy
et al. (1993)applied an aqueous autocatalytic reaction that
produced an irreversibly propagating(reaction-diffusion) front with
characteristics nearly satisfying assumptions (I)-(IV)to simulate
premixed turbulent combustion. Thus these autocatalytic reaction
frontsmay be more justifiably compared to these aforementioned
models, as briefly sum-marized below.The propagtion rates of
reaction-diffusion fronts using aqueous H3As03-IO;
solutions (Hanna et al., 1982) in a nearly isotropic turbulent
flow field have beenmeasured and some of these results are reported
in a recent paper (Shy et al., 1996a).Since these aqueous
propagating fronts have very small exothermicity (typically 1K)and
the solutions are dilute, assumptions (I) and (II) are nearly
satisfied, whereastypically density decreases by a factor of 7 and
kinematic viscosity increases by afactor of 25 across gaseous flame
front. Due to their tiny temperature rise, theaqueous fronts are
unaffected by heat losses (Shy et al., 1993) so that assumption
IIIis nearly satisfied, whereas gaseous flame fronts are strongly
affected by such losses
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FLAMELET AND DISTRIBUTED MODELS 331
(Williams 1985). Thus these aqueous fronts are more appropriate
for comparisonwith the aforementioned combustion theories than are
real gaseous flames. As al-ready discussed in Shy et al. (1996a),
Markstein numbers are probably close to unityfor these chemical
fronts. If Ma = 1 then the influence of flame stretch rate can
beexpressed entirely in terms of a Karlovitz number, such as that
given by Eq. (I).Moreover, since Sc 500 in this aqueous
autocatalytic system whereas Sc I ingases, thus the Karlovitz
number criterion (Eq. 1) reveals that f1amelet behavior maypersist
to much higher values of U'/SL in the aqueous system than in gases.
Further-more, a prerequisite for turbulent combustion study,
experimental or theoreticalmeans, is to obtain (or assume) a
turbulent flow field that is homogeneous andisotropic, because
otherwise the statistical stationarity of flame propagation
cannotbe assured. We have created such a three dimensional
turbulent flow field via a pairof specially-designed,
vertically-vibrated grids in a chemical tank (see Shy et al.,1996
a,b), satisfying assumption (IV). Visualization of turbulent
propagating fronts inthe nearly stationary isotropic turbulent flow
field was via the chemically reacting,LIF technique. In this paper,
we will present the mean reaction progress variable,the variance,
and the pdf of the progress variable which are extracted from
theseplanar LIF front images.Some models of premixed turbulent
combustion employing the f1amelet concept
characterize the structure of the turbulent flame in terms of
the reaction progressvariable (e) modeled either by an algebraic
closure (e.g. the Bray-Moss-Libby, BML,model; see Bray & Libby
1994) or from a transport equation (e.g. Pope 1987). Pope&
Anand (1984) showed theoretically that for f1amelet combustion, the
probabilitydensity function (pdf) of the progress variable was a
double-delta-function distribu-tion, while for distributed
combustion the progress variable pdf showed significantprobability
of partially reacted fluid. At low Ka, Damkohler (1940) proposed
thatthe Huygens propagation model applies, yielding predictions
such as (Yakhot 1988)
(2)
and (Pope & Anand 1984; Bray 1990)
(3)
for large U'/SL where C is a constant (to be discussed later).
At high Ka, Darnkohler(1940) proposed that the distributed reaction
zone (DRZ) model applies (Ronney etal., 1995) in which the front
structure may be disrupted but turbulence influencesST/SL mainly by
increasing diffusive transport inside the broadened front
withoutinfluencing the reaction rate, implying wheresubscripts Tand
L represent turbulent and laminar values. If we assume that
theor-etical relations for Kolmogorov turbulence (Yakhot &
Orzag 1986) may apply atleast for large ReT' then VT/VL 0.061 ReT
and SCT 0.72 whereas SCL 500. TheDRZ model predicts
(4)
-
332 S. S. SHY el al.
Hence, the objectives of this paper which are clearly separate
from prior studies are:(I) to provide experimental data on the
mean, variance and pdf of progress variableand thus compare these
results with the theoretical model of Pope and Anand, (2)
toactually verify whether a steady state ST can exist, and (3) to
compare our experi-mental data on the propagation rates with
theoretical models (Eqs. 2 and 4) andwith other experimental
results using completely different hydrodynamic disturban-ces
(Taylor-Couette and capillary wave flows; see Ronney et al., 1995)
and thusmake the analogy in these cases.The following section
reviews experimental methods used in the study, and' is
followed by a description of the image processing. Both are then
employed toextract the mean reaction progress variable, variance,
and front propagation rates,and conclusion's are offered.
2. EXPERIMENTAL METHODS
2.l. A Region of Nearly Isotropic Turbulence
A flow field which is homogeneous and isotropic over many
integral length scales inall three directions is the most desirable
flow for studying premixed turbulent com-bustion. To fulfill this
requirement, we introduced a vertically-vibrating-grid turbu-lence
that has been commonly used to investigate the entrainment and
mixingmechanisms across a density interface in several geophysical
contexts (e.g. Turner1973; E & Hopfinger 1986). A pair of
specially-designed grids, each composed of tenrectangular bars
yielding 36.9 % solidity, were concurrently vertically
oscillatedthrough the fluid to generate a region of nearly
isotropic turbulence in the coreregion between the two grids (see
Fig. I). This double-grid turbulence generator andits corresponding
flow velocities and statistics are recently described by Shy et
al.,(I996a,b). For completeness, we briefly summarize these results
below.Flow velocities were measured via a two-component laser
velocimeter. A typical
variation of the fluctuating components of flow velocity Ui
(subscript i = 1,2,3 repre-senting the x, y, and z directions)
along the center-line of the tank over the height(H) between the
two grids is displayed in Figure 2a, where w=fS is the
stirringvelocity of the grids. It can be seen that there are two
distinct flow regions: one is anear-grid flow region and the other
is a nearly isotropic region. In the former region,the flow is
roughly homogeneous in horizontal directions (x and y directions)
butinhomogeous in the vertical direction where the mean vertical
velocities (not shown)are greater than the mean horizontal
velocities. In the latter, the mean flow veloc-ities are
essentially zero and r.m.s. fluctuating velocities in all three
directions areapproximately equal (about II 12% of w for H = 10.6
ern), Other measurementpoints are similar. Taken as a whole, a
region of roughly isotropic turbulence ofabout 4 em height can be
generated by concurrently oscillating the two grids with HH = 10.6
ctn.]= I - 8 Hz, and S = 2 cm. The variation among U i in this
region is lessthan IS % for such an arrangement. Also the integral
length scale of turbulence, I"estimated from the Taylor hypothesis
and the autocorrelation coefficient (I, = aU'I,where I is the
turbulent integral time estimated from the Eulerian
autocorrelation
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FLAMELET AND DISTRIBUTED MODELS 333
a be d
H=10.6em
Reactant(Fluorescing)
Motor
M=3em
2DLDASystem
LaserSheet
x
Argon-ionLaser
BeamSplitter
/
z
OF---"y
FIGURE I Schematic diagrams of VGT experimental apparatus. The
argon-ion laser beam wasdirected into the water tank via a beam
splitter and a combination of optical lenses, forming a sheet of250
urn thickness, where the lenses are: a, spherical, focallengthJ=
150 mm; b, cylindrical.j'= - 25.4 mm;c, cylindrical.j'= - 6.35 mm;
d, spherical.j'= 100mm; e, cylindrical.j'= - 100 mm,
respectively.
coefficient and a =J8j;; (Abdel-Gayed et al., 1987», is found to
be nearly a constantabout 0.3 cm, as shown in Figure 2b. The
constant I, occurs because the integraltime scale is found to be
inversely proportional to w = is while the overall turbu-lence
intensity is proportional to w. In this study, the Reynolds number
based on thegrid mesh size was varied from 600 to 4,800 and the
ratio of U'jSL may range fromabout 20 to about 3,000 with an
emphasis on small and moderate values of u'jS/.( < 400)
corresponding to the flamelet mode.
2.2. Chemical Apparatus
As in previous studies (Shy et al., 1993; 1996a), the chemical
system we employed isthe aq ueous arsenous acid-iodate reaction.
This reaction system consists of the
-
334
Lower-GridLocation
1
0.75
0.5
0.25
o
-0.25
-0.5
S. S. SHY et al.
Upper-Grid-O--u 1/W Location--+-u2/w---e -u 3/w 1 I(a)D
-u1u3/w·
-.-.- ----
-I -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1z/0.5H
I II I" (b)---J t -. ::d).... ___.._______.__ "k- I -.. I-
--
1:-·OoCO d-_.-_._--_._- -_.- ._._- • .-..----- --"_.*- _.-----
---f-.--Q.-----I :
i i .
0.4
0.3
0.2
0.1
o-0.8 -0.6 -0.4 -0.2 0 0.2
z/0.5H0.4 0.6 0.8
FIGURE 2 Typical characteristics of VGT apparatus. (a) r.m.s, of
fluctuating velocity components(",.",. "Jl and the turbulent shear
stress ( -", "J ) along the centerline of the tank. (b) Variation
of theintegral length scale with the distance between the two grids
where the solid and empty symbols representthe vertical and
horizontal components, respectively. Both grids oscillated at f =6
Hz, S= 2 em, andH= 10.6em.
-
FLAMELET AND DISTRIBUTED MODELS 335
Dushman reaction (10; + 5 1- + 6 H + --+ 3 Iz+ 3HzOJ followed by
the more rapidRoebuck reaction (H3As03 + Iz+ HzO --+ H3As04 + 2 1-
+ 2 H+). If the arsenousacid is in stoichiometric excess ([H 3As03J
O > 3[10;Jo; subscript 0 represents theinitial concentration
value), the net reaction is Dushman reaction + 3 • Roebuckreaction
or 3 H3As03 + 10; --+ 3 H3As04 + 1-. Such a reaction is
autocatalytic iniodide (I ") because it catalyses the Dushman
reaction. By electrochemically initiat-ing this reaction system,
propagating fronts with constant SL can be obtained(Hanna et al.,
1982).All solutions were prepared using reagent-grade chemicals and
deionized water.
In this work experiments were conducted at conditions
.corresponding to values ofKa ranging from about 0.1 to 300, with
an emphasis on values of Ka < 10 becausethe apparent limit for
flame1et behavior in these aqueous autocatalytic fronts isKa 5 (Shy
et al., 1993, 1996a; Ronney et al., 1995). For the aqueous
reactionsystem, the conventional definition of Karlovitz number
(Ka) requires some modifi-cation. As discussed by Shy et al.,
(1993), in general Ka is defined as the ratio of themean turbulent
strain rate (u'/I,)Rej!2 (which can readily be shown to be equal to
thestrain rate at the Kolmogorov scale) to the mean chemical
reaction rate - SUD.This ratio can be rearranged to obtain Ka
(U'/SJ2 ReT 1{2 Sc- I • For gaseousflames Sc 1 and thus the Schmidt
number has been ignored in prior definitions ofKa. For consistency
with the definition proposed by the Leeds group, we incorpor-ate a
factor of 0.157 in the definition of Ka (Eq. 1). Alternatively, Ka
is sometimesdefined in terms of the ratio of the laminar flame
thickness - D/SL to the Kol-mogorov length scale I. -I, ReT3{4
which can be expressed as (U'/SL)ReT 1{4 Sc- 1.Thus for gaseous
flames with Sc 1, the two definitions are equivalent if Ka
isdefined as the square of the ratio of the laminar flame thickness
to the Kolmogorovlength scale. For aqueous systems with Sc» 1, the
definition based on length scalesresults in a much lower Ka, by a
factor of Sc- I, than the definition based on the ratioof strain
rate to chemical rate. In our studies of aqueous front propagation
we havechosen to employ the definition based on rates because it
would seem that flamelet-like behavior would cease if either the
mean strain rate were higher than the meanchemical rate or the
flame thickness were larger than the Kolmogorov length scale.The
front thickness, 0, can be estimated (Williams 1985) as D/SL with D
2.0 X
10- 5 cmz/s for 1- in water (Hanna et al., 1982). Reactive
solutions of [H3As03Jo =0.1 0.15 M and [IO;Jo= 0.005 - 0.03 M, at a
pH value of 6.8 and an initialtemperature of 25°C, were used for
these experiments. This arrangement providedvalues of SL ranging
from about 7 to 75 X 10- 3 rnrn/s. Thus the estimated 0 mayrange
from about 27 to 286 Jlm which was generally smaller than the laser
sheetthickness urn). As sketched in Figure 1, a 5 mm argon-ion
laser beam(Coherent Innova 90-5W) was directed into the tank by
means of two speciallycoated mirrors and a combination of spherical
and cylindrical lenses, to ensure thatthere was little change in
the thickness of the laser sheet with distance from thelenses,
using similar optical lenses combination as Prasad and Sreenivasan
(1990).Thus, the main limitation on the image resolution is
probably the laser sheet thick-ness rather than the chemical front
thickness.The propagating front was detected by chemically
reacting, laser-induced fluores-
cence (LIF) technique. Initially, a small amount of disodium
fluorescein (typically
-
336 S. S. SHYer al.
less than 10- 5 M) was added and mixed in the autocatalytic
solution. When excitedby an argon-ion laser sheet, the dye
fluoresced if the pH value of the solution wasgreater than 4,
otherwise no fluorescence occurred. Thus the reactants (pH
6.8)fluoresced, and upon autocatalytic reaction, the bright
reactants were converted intothe dark products (pH 2.3) (see Fig.
1). Note that the fluorescence transition forthe laser dye was
occurred on a very short time scale (on the order of tens
ofnanoseconds; Koochesfahani 1984).
2.3. Experimental Procedures
A run began by initiating reactants at the top surface of
solution using potentialdifferences between Pt electrodes. The
autocatalytic reaction. then spread uniformlyover the entire
surface of the solution and developed into a stable
downwardpropagating front with a constant speed. With sufficient
products generated, thetwin grids were then vertically oscillated
in opposite phases. The front propagateddownward and converted
reactants into products. In the region of interest, i.e. theregion
of nearly isotropic turbulence, we recorded the evolution and
spatial struc-tures of the turbulent propagating front with a super
VHS video system and a motordriven 35 mm camera.
2.4. Image Processing
The L1F images were digitized into 512 x 256 pixel arrays, with
each pixel beingassigned a value of brightness between 0 and 255
(8-bit). The analyzed zone corre-sponded to a rectangular region of
12 em long and 6 em high which was centered atthe core region
between the two grids. Each pixel had a physical space of 234 11m
x234 11m. Finer image resolution would not have been useful because
the laser sheetthickness was about 250 11m.Figure 3a displays the
brightness profile along a line of pixels across the mean
position of the wrinkled front at conditions of U'/SL 47 and Ka
0.2. It can beseen (Fig. 3a) that the interface between reactants
(high brightness) and products(low brightness) is quite sharp. At
column 195 (Fig. 3a), there is a discontinuityindicating that the
very sharp front is perpendicular to the line of pixels. Such
a.f1amelet-like behaviour of the front can be sustained for values
of Ka up to at least 5in this aqueous autocatalytic system (Ronney
et al., 1995; Shy et al., 1996a). Thehistogram of pixel brightness
for an entire image corresponding to the case of Figure3a is shown
in Figure 3b. The images are then digitized with c = 0 for
reactants andc = I for products by selecting a digitized threshold
value which in flamelet-likecases can be easily determined without
any ambiguity. However, we agree that theappearance of sharp fronts
based on the LIF image is a neccessary but not sufficientcondition
for f1amelet behavior. In comparison, Figure 4 shows the result for
ahigher value of Ka where U'/Sl. 662). Clearly, the interface is no
longersharp (i.e. front structures reminescent of distributed
combustion are observed),causing some doubts in choosing the
digitized threshold value which in this case isselected as the
brightness value at the well of the histogram (= 64; Fig. 4b).
Bythresholding, pixels with values of brightness less than 64 are
equal to unity (c = I)
-
FLAMELET AND DISTRIBUTED MODELS 339
corresponding to products while those greater than or equal to
64 are zero (c = 0)corresponding to reactants.However, this
criterion is somewhat arbitrary (the sensitivity of the threshold
value
will be discussed in the next section), and so far larger Ka it
is expected that models ofturbulent combustion developed assuming
flamelet behaviour may not apply.
3. RESULTS AND DISCUSSION
3.1. Turbulent Propagating Fronts After Thresholding
As shown in Figure 3b and 4b, the probability distribution of
the light intensity (thehistogram of pixel brightness) for these
LIF images, which looks similar to theprobability density function
of progress variables, exhibits a nearly double-deltafunction
distribution for values of Ka < 5 while for Ka 18 there is
significantprobability of intermediate distribution (partially
reacted fluid). These results sup-port the theoretical description
of flamelet and distribution combustion proposed byPope and Anand
(1984) and confirmed experimentally by Yoshida (1990).
However,since the light intensity may not be simply related to the
progess variable, the resultsfor high Ka must be viewed at most
qualitatively and with caution.Four instantaneous, 2-D images
obtained after thresholding procedures for vary-
ing values of U'/SL and/or Ka are shown in Figure 5. It can be
seen that thepropagating front becomes more and more wrinkled as Ka
increases. At Ka 2, it ispossible to observe some islands or
pockets of unreacted material (the reactant),where the propagating
front still remains sharp. As Ka increases further (above 10,shown
in Fig. 5d), many islands of reactants as well as pockets of
products can beobserved in the 2-D image. This seems to be
consistent with the observation (Shyet al., 1993) that only at high
Ka does local quenching occur and permit islands ofproducts to
form. As explained by Ronney (1995), only when local front
quenchoccurs can islands of products occur. However, in order to
truly confirm the aboveisland formation, observation images in the
third direction have to be obtained. Thisis currently being studied
(by the first arthor) using a 3-D successive planar LIFimaging of a
synchronized raster swept laser beam, combined with a very fast
dataacquisition system, and these results will be reported
elsewhere in the near future.
3.2. Mean Reaction Progress Variable and Variance
Figure 6 displays the profiles of the mean and variance of the
progress variable c forfour different values of Ka and U'/SL
corresponding to those in Figure 5. Here thecoordinate z :is
measured from the top of the binarized image (Fig. 5) and
nor-malized by I, (Fig. 2b). For all values of Ka studied here, the
fronts propagate to theright into reactants (c = 0), leaving
products (c = 1) behind it. The variance (c - C)2 iszero outside
these propagating fronts where c is either zero (reactants) or
unity(products); the overbar represents the ensemble averaged. The
variance reaches apeak close to where c= 0.5. If the front were
infinitely thin, then the maximum valueof the variance would be
0.25 at c=0.5 (Fig. 6).
-
FLAMELET AND DISTRIBUTED MODELS 341
1.0 O.S... ,0.9 l- • • c
Q,l 0 (c - c)'i •0.8 • 0.4'[ij;> 0.7 -.•III •0.6 - • - 0.3
Q,lC!I §£ O.S £= '[ije 0.4 f- 0.2 ;>'.Q
0 00.3 f- 0 0--•§ 0.2 f- 0.1Q,l ·00
\0.1 I- (a) Ka""O.2 0it0.0 0.0
0 2 4 6 8 10 12 14 16 18 20
z/lt
1.0 O.S
0.9 • cQ,l 0 (c - C)Z
i 0.8 • 0.4'[ij •;> 0.7 • ..III ,III
0.6 A 0.3 Q,lC!I
-
342 S.S. SHYer al.
1.0 0.5
0.9 • c0 (c· c)'0.8 ..
'!ij •;> 0.7 •IIIIII ..0.6 • "'"£ 0.5 a'!ij=Q 0.4 eo ;>al
0 -ell 0.3 crPCI:a 0.2
0.1 (c) Ka'" 2.00.0 0.00 2 4 6 8 10 12 14 16 18 20
z II,
1.0 , I , , O.S- C0.9 I- 0 (c • C)2-'§ 0.8 - 0.4'!ij;> 0.7
I-III -III!::! 0.6 l- • 0.3 "'"£ =O.S I- Jfi (\I'!ij=Q 0.4 I- - 0.2
;>ali! 0
0.3 l-
t --
-
FLAMELET AND DISTRIBUTED MODELS 343
As can be partly seen in Figure 6, the span of the mean and
variance of thereaction progress variable, which can be viewed as
an indication of the flame brushthickness (lFO) defined as the
width of the variance at 0.125 (one half of the maximumvalue of the
variance), increases rapidly from about 1.6 integral length scale
ofturbulence (I,) to about 4.8 I, as Ka increases from 0.2 to 4.
When Ka increases up to5 or above, If b only increases slightly and
reaches to about 5.2 I, at Ka "" 18 (Fig. 6d).For the f1amelet mode
(Ka < 5), the profiles of cand (c -cf look roughly symmetricwith
respect to the location where the variance is at its maximum value.
However,for large values of Ka (such as Ka e 18; Fig. 6d), the
profiles of c and (c _E)2 departfrom the symmetry, rapidly approach
their boundary values on the product side(c = I and (c - E)2 =0),
and act slowly on the reactant side ( c= 0 and (c _E)2 = 0).This
result is in good agreement with the behaviour proposed
theoretically by Popeand Anand (1984). Again we note that the
current aqueous chemical system is not anArrhenius reaction system
with high activation energy, so that results for highervalues of Ka
(> 5) have to viewed at most qualitatively and with caution.
Also c(I - E) is essentially equal to the variance, (c _E)2, within
experimental uncertainties,because the front interface is assumed
to be infinitely thin. Thus progress variablevariance against mean
reveals a parabolic curve with the maximum value of thevariance (c
- E)2 =0.25 at c= 0.5, in excellent agreement with the theoretical
modelsproposed by Pope and Anand (1984).
o Threshold = 60• Threshold = 64t> Threshold = 68
2018161410 12z II,
8642
1.0
0.9CIl
0.8'!ij 0.7....tiltil 0.6
£. 0.5= 0.4Q;j 0.3
a 0.2CIl::; 0.1
0.00
FIGURE 7 Mean progress variable against normalized distance for
three different thresholding valuesin mode similar to distributed
behavior (Ka '" 18). indicating the sensitivity of thresholding
values.
-
344 S. S. SHY et al.
3.3. The Sensitivity of the Threshold Value
In order to test the sensitivity of these results to threshold
values, the effect ofsmaller or larger values of the brightness
threshold were tested (60 or 68) (see Fig.4b). Figure 7 displays
the mean progress variable against the normalized distance forthree
different threshold values (60, 64 and 68; see Fig. 4b). It is
found that changingthe threshold value; either close to the
reactant side or close to the product side,does not alter the
primary shape of c, indicating the insensitivity of the results to
thethreshold value even for the value of Ka as large as 18.
3.4. Verification of a Steady-State Sr
Typical digitized images of the turbulent front propagation
through the nearlyhomogeneous isotropic turbulence region, the
region in which Sr was inferred, areshown in Figure 8 for Ka 0.2.
In order to verify whether a steady state Sr can existin this
chemical system, we measured a mean product creation height (h)
which wasthe products (dark) area per unit tank width measured from
the middle plane of theupper grid as a function of time using a
simple binarized area counting program. Srwas then defined as the
mean vertical propagation rate of product (dh/dt), similar tothe
conventional definition of Sn since Sr multiplied by the
cross-sectional area ofthe tank represented the volumetric rate of
product creation. In Figure 8, to is thetime period measured from
starting the vibrating grids to the mean front justreaching the
upper boundary of nearly isotropic turbulence region. Each
binarized
o o o o o o - The middle position of the upper grid2.3 -L
Regionof
Interest(b) 9.37 sL ----.J'--r '-- _8.3
z(cm)
i (c) 17.46 s
--r4cm
-L (d) 24.27 sFIGURE 8 Typical binarized image sequence of
propagating front in nearly-isotropic turbulence forKa 0.2 and
.'ISL 47. Field of view 60 mm x 120mm.
-
FLAMELET AND DISTRIBUTED MODELS 345
image was cut in two from the center of the image and estimated
ST separately totest the symmetry of front propagation. Figure 9
shows that an essentially steadystate ST does exist in this
chemical system and that the front propagation is almostsymmetric
with respect to the centerline of the chemical tank.
3.5. Front Propagation Rates
The effect of U'/SL on ST/SLwas measured for a 3-decade range of
Ka, correspond-ing to values of U'/SL between 38 to 2849. It was
found that at moderate Ka ( < 5),the determination of ST is
quite accurate because the front interface is very sharp,but at
higher Ka (Ka > 10) the front interface becomes diffuse and
"fuzzy", suggest-ing a transition from flamelet to distributed
behavior. Typically at large values ofKa, there are two possible
binarized threshold values to be chosen, corresponding tothe upper
and lower boundaries of the fuzzy interface. The variation of ST
using theabove two different threshold values are found to be no
more than 5 %. Thisindicates that the average thickness of the
broadened front may remain roughlyconstant at a given ReT and Ka
(Shy et al., 1996a).Figure lOa shows values of ST/Sl. = UT as a
function of U'/SL = U and compares
experimental results for Ka < 5 with analogous theoretical
results (Pope & Anand
121110\I8567t- ... (sec)
-- Right- - - - Left
ST=0.7564 cmls"" "",dh
ST = dt =0.1067 cmls----- -_ • '/__A - - - - • - -
ST =0.0576 cmls
\I
The Middle Positionl/oftheUpperGrid
.........I-""-L.....I......I...........I.-"""""""'''''''''''''''''..I...-'''''''...................................o
1 Z 3 4
"',6: u'/SL"" 47; Ka "" 0.210
...........,.......,..........."""T..........-""T"".....-I .,0:
u'ISL "" 143; Ka "" 2.0
_,0: u'/SL"" 662; Ka"" 18
FIGURE 9 The time evolution of the product creation height,
which determines the front propagationrate, ST'
-
346 S. S. SHY eral.
1984; Bray 1990;Yakhot 1988; Shy et al., 1993).Two features of
these data should benoted. First, the experimental data on UT are
in good agreement with Eq. (2), theprediction of Yak hot's
renormalization group theory. This agreement is very goodfor
mixtures with higher SL at lower U'/SL, but less satisfactory with
lower Sf. athigher U'/SL' suggesting a strain rate (Karlovitz
number) effect on Sv Second,measurements of UT are much higher than
that predicted by Shy et al., (1993) forfront propagation through
an array of singlescale vortices generated by a Taylor-Couette
flow, while much lower than those by the presumed-pdf method of
Bray(1990) and the joint-pdf theory of Pope and Anand (1984). The
latter point isprobably due to the fact that these theories
incorporated certain assumptions re-garding the statistical
properties of the flow or some empirical constants taken fromgas
combustion experiments which naturally exhibited specific velocity
spectra. Asvalues of Ka increase up to 10, values of U T tend to
bend down from the Yak hot'sprediction. Such a departure of the
linearity of UT as a function .of U is even morepronounced for
Ka> 10 (the small diagram in Fig. lOa; see Shy et al.,
1996a).Figure lOb displays the ratio of experimental values of U T
to the theoretical predic-tions of Eq. (2) as a function of Ka,
Values of U T for Ka < 10 are in good agreementwith the
predictions of Eq. (2), and the ratio of experimental to
theoretical valueshas a mean of 1.13 and rms deviation of 0.36.
This good agreement at low Ka(f1amelet mode) is probably because
Eq. (2) is based on an exact equation forHuygens propagation model
(Kerstein et aI., 1988) and does not require empirical
300--...l
25000-'-'
200eu=:cQ 150.-....euOileuCl. 100QIt....c 50Q"'"r..
00 50 100 150 200 250 300 350 400
Turbulent Intensity (u'/SL)FIGURE 10
-
FLAMELET AND DISTRIBUTED MODELS 347
I'I •
I •!
100(Ka)
101Karlovitz number
o
(b) I
_ .. _I •• vo· ....
I I.
1
0 I tftI Ii 1I I
I i
1
0.10.1
10
110 100
Karlovitz number (Ka)FIGURE 10 (a) Effect of turbulence
intensity on the simulated turbulent burning velocity and
compari-son with predictions of some flamelet models of turbulent
combustion and a model of front propagationin a one-scale flow (Shy
et al., 1993). These dark circles represent previous results (Shy
et al., 1995a) whilethese white circles are the new results. (b)
Comparison of measured Sr/SL values to a Huygens propaga-tion model
(Eq. 2). (c)Comparison of measured Sr/SL values to a distributed
reaction zone model (Eq. 4).
-
348 S. S. SHY et al.
constants extracted from gaseous combustive experiments that may
not satisfy as-sumptions (l)-(IV). In an independent, separated
study using completely differentstirring methods, Ronney et al.,
(1995) found the similar results, suggesting thatvelocity spectra
may have only a weak effect on U l' as long as the flow has
broadenough spectra of flow scales. While it is not yet known how
broad a range of scalesis required for scale-independence, Ronney
(1995) has proposed a speculative cri-terion based on the slope of
the conventional kinetic energy density vs. wavenumberplot. It was
estimated that a rather steep slope of - 4, compared to - 5/3
forclassical Kolmogorov turbulence, would be required before the
range of scaleswould be narrow enough that 51'would exhibit an
appreciable scale-dependence. Todate no experimental or numerical
studies have been performed that would providea test of this
proposition.Figure 10c shows the ratio of experimental data of U r
to the distributed reaction
zone (DRZ) prediction (Eq. 4) for Ka > 10. Experimental data
are roughly consistentwith DRZ predictions at Ka> 10 except a
factor of 5 higher; the ratio of experimen-tal values of UT to
theoretical predictions has a mean of 5.14 and rms deviation
0.68.In comparison, there is no evidence from gaseous combustible
flame fronts thatEq, (4) is valid. This might be due to the factor
that Eq. (4) was derived by assumingthat the chemical reaction is
not influenced by turbulence. On the other hand, localreactions of
gaseous flames with large activation energy are sensitive to small
fluctu-ations in local temperature due to turbulence which may
result in a largely non-linear change in the local reaction rate
and thus affect its mean value (Ronney et al.,1995). Such a strong
nonlinearity is absent in the aqueous H3As03-IO; system(Hanna et
al., 1982).Because of the absence of heat loss influences and the
lack of strong nonlinearities
in reaction rates along together with constant density and large
ranges of Ka and Uthat we are able to employ, our experiments might
be the first that could be directlycompared to the DRZ model
proposed by Damkohler (1940). In different turbulentflows (both
TaylorCouette and capillary wave flows), Ronney et al., (1995)
foundthat their experimental results were in good agreement with
Eq. (4). Thus the factorof 5 discrepancy between our high-Ka
experimental data and the DRZ model in UT(Fig. IOc) is surprised
and the reason for such a discrepancy is not clear at thismoment.
However, we anticipate three possibilities: (I) The DRZ model, like
allmodels, is speculative, especially the factor 6.5 in Eq. (4)
that was obtained from theassumptions of vl'/vL 0.061 Re; and SCT
0.72 (see Yakhot and Orszag 1986). (2)Due to the practical limit of
the size of the tank, the turbulent Reynolds numberbased on the
integral length scale of turbulence may be not large enough.
(3)While I,was measured in the same manner as employed in prior
studies of turbulentpremixed flames (see section 2.1), the
resulting value of I, 0.3 cm is considerablysmaller than the
maximum size of flame wrinkling ( 3 cm) seen in Figure 5.
Somestudies in gaseous flames (Mantzaras et al., 1988; North and
Santavicca 1990) andautocatalytic fronts (Haslam and Ronney 1995)
have suggested that I, is the outercutoff scale for front
wrinkling. If I, 3 em were tentatively used to estimate ReT andthus
51'/5" in the DRZ regime, the agreement between model and
experiment wouldbe considerably improved at which the mean of the
ratio of experimental totheoretical values would drop from 5.14 to
1.63. Detailed measurement of front
-
FLAMELET AND DISTRIBUTED MODELS 349
wrinkling scales relative to turbulence scales is deferred to a
future study. Neverthe-less, Figure IDe is presented here in the
hope that it may serve a constructive purposeby stimulating
additional research and discussion.
CONCLUDING REMARKS
Experiments of these aqueous autocatalytic fronts propagating in
a nearly isotropicturbulent flow field have been conducted to
obtain velocity and front surface prop-erty data, thereby enabling
comparison with flamelet models. Our concluding re-marks are: (I)
The present work supports the theoretical view of flamelet and
distrib-uted combustion proposed by Pope and Anand based on a mean
progress variable;(2) For flamelet mode of combustion, our results
indicate that the velocity spectrumhas an effect on ST/SL' even at
a fixed U'/SL' and values of ST/SL are in goodagreement with
Yakhot's prediction; (3)Values of U'/SL can be obtained much
higherthan those attainable in gaseous combustion experiments,
possibly suggesting theresilience of wrinkled flamelets to
turbulence and the importance of heat loss.
ACKNOWLEDGEMENTS
This research was supported by National Science Council. Taiwan,
R.O.C., under Grant No. 83-0401-E-008-056 and 84-2212-E-008-023.
P.D.R. was supported by the NASA Lewis Research Center under
GrantNAG3-1523.
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