Sci. Tech.• 1996. Vols. 113-114, pp. 329-350Reprints available directly from the publisherPhotocopying permitted by license only

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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

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

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

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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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