Energy & Fuels 1993,7, 827-834
827
Binary Scalar Mixing and Reaction in Homogeneous Turbulence: Some Linear Eddy Model Results S. H. F r a n k e l , ? C. K. Madnia, P. A. McMurtry,z and P. Givi' Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo,Buffalo,New York 14260-4400 Received April 8, 1993. Revised Manuscript Received September 4, 1993"
The linear eddy model (LEM) of Kerstein (Kerstein, A. R. Combust. Sci. Technol. 1988; 60, 391-421; J. Fluid Mech. 1991,231,361-394; Phys. Fluids A 1991,3 (5), 1110-1114; J. Fluid Mech. 1992,240,289-313) is used to simulate the mechanism of scalar mixing from an initial binary state
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in incompressible, homogeneous turbulence. The simulated results are used to measure the limiting rate of mean reactant conversion in a chemical reaction of the type F + rO (1 + r)products under isothermal and nonpremixed conditions. The objective of the simulations is to assess the performance of the closed-form analytical expressions obtained by Madnia et al. (Madnia, C. K.; Frankel, S. H.; Givi, P. Theor. Comput. Fluid Dyn. 1992, 4, 79-93) based on the amplitude mapping closure (Kraichnan, R. H. Bull. Am. Phys. SOC.1989,34,2298; Chen, H.; Chen, S.; Kraichnan, R. H. Phys. Rev. Lett. 1989,63 (24), 2657-2660; Pope, S. B. Theor. Comput. Fluid Dyn. 1991,2, 255-270) for the evaluation of the mean reactant conversion rate. This assessment is made for various flow conditions with different asymptotic statistical behavior.
by Miller et al.,15 the most serious of these is the incapability of the model to account for the migration of In a recent work, Madnia et aL5 provide closed-form scalar bounds as mixing proceeds. This problem manifesta analytical expressions for estimating the limiting rate of itself in the conditional statistics of the scalar variable, mean reactant conversion in a binary chemical reaction of namely, the conditional expected dissipation and/or the (l+r)product in homogeneous isothe type F + rO conditional expected diffusion. Furthermore, due to thermal incompressible turbulent flows. These expresobvious limitations of DNS for simulatinghighly turbulent sions are obtained by means of a single-point probability flows, other means of assessing the performance of the density function (PDF) method based on the amplitude closure are required. In this work, our objective is to mapping closure (AMC).- The performance of the model, further evaluate the performance of the results of Madnia as assessed by comparison with the results of direct et aL5 as a complement to those reported in refs 5 and 9. numerical simulations (DNS), has been very encouraging, For this purpose, we have made use of results generated and the results seem to compare with DNS data better than all the previously available closures in the l i t e r a t ~ r e . ~ by the linear eddy model (LEM) originated by Kerstein.' The capabilities of this model have been demonstrated in Moreover, the simplicity of the final analytical expressions a number of test cases,14J6-18 and based on these studies makes them very attractive for practical applications in the use of the model is well justified for the evaluation plug flow reactors and batch mixers.'&14 purposes intended here. Despite this demonstrated superiority, there are some drawbacks associated with AMC. As discussed in detail 1. Introduction
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2. Formulation
Current address: School of Mechanical Engineering, Purdue University, Wwt Lafayette, IN 47907-1077. t Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112.0 Abstract published in Advance ACS Abstracts, October 15, 1993. (1)Kerstein, A. R. Combust. Sci. Technol. 1988,60,391-421. (2)Kerstein, A. R.J. Fluid Mech. 1991,231,361-394. (3)Kerstein, A. R. Phys. Fluids A 1991,3(5),1110-1114. (4)Keretein, A. R. J. Fluid Mech. 1992,240,289-313. (5)Madnia, C.K.; Frankel, S. H.; Givi, P. Theor. Comput. Fluid Dyn. 1992,4,79-93. (6)Kraichnan, R. H. Bull. Am. Phys. 1989,34,2298. (7) Chen,H.; Chen, S.; Kraichnan, R. H.Phys.Reu.Lett. 1989,63 (24), 26.57-2fiIin. - - - . -- - -. (8)Pope, S.B. Theor. Comput. Fluid Dyn. 1991,2,255-270. (9)Frankel, S.H.; Madnia, C. K.; Givi, P. AIChE J. 1993,39, (5), 899-903. (IO) Frankel, S. H.; Jiang, T.-L.; Givi, P. AIChE J. 1992,38(41,535543. (11)Toor, H. L. In Turbulence in Mixing Operations; Brodkey, R. S., Ed.;Academic Press: New York, NY, 1975;pp 123-166. (12)Hill, J. C. Annu. Rev. Fluid Mech. 1976,8,135-161. (13)Givi, P. B o g . Energy Combust. Sci. 1989,15,1-107. (14)McMurtry, P. A.; Givi, P. InNumen'cal Approaches t o Combustion Modeling; Oran, E. S., Boris, J. P., Eds.; Volume 135 of Progress in Astronautics and Aeronautics; AIAA Publishing Co.: Washington, DC, 1991;chapter 9, pp 257-303. t
The reacting system under consideration is a homogeneous turbulent flow field under the influence of an irreversible binary reaction of the type F + r0 (1 + rlproduct with initially segregated reactants (F and 0). The turbulence field is assumed statistically stationary, homogeneous, and isothermal, and all the chemical species are assumed to have identical and constant thermodynamic properties. This system provides a good model for dilute reacting systems in typical mixing-controlled plug flow reactors.11J2J9-21 In the limit of infinitely fast chemistry (or chemical equilibrium in this case), the statistical behavior of the
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(15)Miller, R.S.;Frankel, S.H.; Madnia, C. K.; Givi,P. Combust. Sci. Technol. 1993,91 (1-3),21-52. (16)Kerstein, A. R. Combust. Flame 1989,75, 397-413. (17)Kerstein, A. R. J. Fluid Mech. 1990,216,411-435. (18)Kerstein, A. R. Combust. Sci. Technol. 1992,81,75-96. (19)Toor, H. L. AIChE J. 1962,8,70-78. (20)Bilger,R. W.In TurbulentReactingFlows; Libby,P. A., Williams, F. A., Eds.; Springer-Verlag: Heidelberg, 1980; chapter 3, pp 66-113. (21)Brodkey, R. S. Chem. Eng. Commun. 1981,8,1-23.
OS87-0624/93/25O7-0827$04.00/0 0 1993 American Chemical Society
828 Energy & Fuels, Vol. 7, No. 6,1993
Frankel et al.
reacting scalar variables can be related to the statistics of an appropriate conserved Shvab-Zeldovich mixture fraction, This mixture fraction can be normalized in such a way to yield values of unity in the fuel F stream and zero in the oxidizer 0 stream. For the purpose of statistical treatment, we define P ~ ( 4 , t )Po(#,t), , and P J ( @ , ~respectively, ), as the marginal PDFs of the mass fraction of F, the mass fraction of 0, and the ShvabZeldovich variable J. For initially segregated reactants with no fuel in the oxidizer stream and vice versa, the initial conditions for the marginal PDFs of the mass fraction of the two reactants are given by
is described in refs 5 and 24. Here, the final results are presented for the mean values of the two reactants:
J.20922
In these equations, c is related to the stoichiometric coefficient, Here, Fi and Oi denote the initial mass fractions of the two species in the two feeds, and WF and Wo represent the relative weights of the reactants at the initial time (i.e., the area ratios at the inlet of a plug flow reactor). With the normalized value of the mass fractions equal to unity at the feeds, i.e., Fi = Oi = 1,the stoichiometric value of the Shvab-Zeldovich variable, Jst, is determined from the parameter r. With the assumption of an infinitely fast chemistry, the marginal PDFs of the reactants' concentrations are related to the frequency of the ShvabZeldovich ~ a r i a b l e . Therefore, ~ ~ * ~ ~ the temporal variation of the statistics of the species field are easily determined if the temporal evolution of P ~ ( 4 , tis) known (from here on the subscript J i s dropped). The AMC provides a means of describingthis evolution. The implementation of AMC involves a mapping of the random field of interest 4 to a stationary Gaussian reference field 40,via a transformation 4 = x(@o,t). Once this relation is established, the PDF, P(4),is related to that of a Gaussian distribution. For the case considered here, the PDF of the Shvab-Zeldovich variable adopts the form5
In this equation, 7 is a nondimensionalized time and its relation with the physical time cannot be determined in the context of single-point description.8 The variable $J* provides a measure of the initial asymmetry of the PDF: $* = fierf'(1- 2 ( J ) )
(3)
where "err denotes the error function, and ( ) denotes the ensemble average. With this notation, (J)= WF. Also, G(7) = (exp(27) - l)'/',
1 fiG'
(7) and
y2 = y2
1 +2a2
b c y, = r2acy z ---a2 a Here, I l - 2 and i9-1 are the parabolic cylinder functions of order -2 and -1, re~pectively.~~ Due to nice properties of the degenerate hypergeometric functions, many of the interesting features of eqs 5 and 6 can be depicted. For practical applications in stoichiometricplug flow reactors, the equations simplify considerably. For the case of an initially symmetric PDF around the mean value (Le., ( J ) = Jat= l/z) both parameters b and c are zero. Therefore, the first terms on the right-hand side (RHS) of eqs 5 and 6drop. Knowing2)-2(0)= D-1(0)= 1,the remainingterms yield
( F ) ( 7 ) (0)(7) -(F)(O)- zz = zJO
1
dy 2y2 + l/a2 2 arctan G ( T ) 1(9)
1
-
7r
This equation can be conveniently expressed in terms of the "Intensity of Segregation,"26defined as the normalized variance of the Shvab-Zeldovich variable (a2). For AMC with an initial symmetricPDF, this intensity is given by2"JO
a(7) = -
b(7) =
+*(l
+ G2)'12 (4)
f i G
With a combination of eqs 1 and 2 all the pertinent single-pointstatistics of the reacting field are determined. The mathematical procedure for evaluating higher order statistical quantities based on the AMC generated PDF (22) Williams, F. A. Combustion Theory, 2nd ed., The Benjamin/ Cummings Publishing Company: Menlo Park, CA, 1985. (23) Kosaly, G.; Givi, P. Combust. Flame 1987, 70, 101-118.
A combination of eqs 9 and 10 yields (24)Frankel, S. H. Ph.D. Thesis, Department of Mechanical and AeroepaceEngineering,State Universityof New Yorkat Buffalo,Buffalo, NY, 1993. (25)Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functionsand Formula, Graphs,and Mathematical Tables;Government Printing Ofice: Washington,DC, 1972. (26) Brodkey, R. S., Ed. Turbulence in Miring Operation;Academic Press: New York, NY, 1975. (27) Jiang, T.-L.; Gao, F.; Givi, P. Phys. Fluids A 1992,4 (5), 10281035.
Binary Scalar Mixing in Homogeneous Turbulence
Energy & Fuels, Vol. 7, No. 6,1993 829
The simplicityof this equation is noteworthy and provides a closed-form expression for the mean reaction conversion rate in the limit of infinitely fast chemistry. This equation is to be compared with those suggested in earlier contributions. For example, the model based on Toor’s hypothesis’l
where F denotes the cumulative distribution function (CDW F ( 4 , t ) = J:P(+,t) d+
-= ( F )( t )
&1/2
(12)
(20)
and the 3E model of Dutta and Tarbel128
The family of exponential PDF’s has been useful in analyzing the problem of binary scalar These PDFs are expressed as
In a nonstoichiometric mixture it is not possible to make the AMC results any simpler, and numerical integrations of eqs 5-8 are required. In this regard it must be noted that the 3E model and the results based on Toor’s hypothesis are only valid for stoichiometric mixtures.
where q = 2 corresponds to a Gaussian PDF, and q = 1 implies a Laplace (exponential) density. Considering PDFs with a mean (4) and a variance u2, eqs 18 and 19 can be used to evaluate the conditionalstatistics. Straightr forward manipulations of these equations show that
( F )(0)
3. Evolution of the PDF
In the problem considered here the essential physics are embedded in the PDF of the Shvab-Zeldovichvariable. Therefore, an understanding of its basic transport is essential. For an incompressible turbulence, the evolution of the PDF of a conserved scalar variable, P(&,t), 4 E [&,4uI is governed by29
for the Gaussian PDF, and
where E represents the conditional expected value of the scalar dissipation. Equation 14 can alternatively be expressed as
where D denotes the conditional expected value of the scalar diffusion and is related to E through
At the single-point level neither the conditional mean dissipation nor the conditional mean diffusion are known, nor are their unconditional mean values including the total dissipation, t:
- J:@($,t)
o(4,t)d4 (17)
However, given the PDF it is possible to determine the conditional statistics. Following the procedure outlined in ref 15 we write (28)Dutta, A.;Tarbell, J. M. AIChE J. 1 9 8 9 , s (12),2013-2027. (29)Pope, S. B. h o g . Energy Combust. Sci. 1985,11, 119-192.
for the Laplace PDF. It is noted that the values of the conditional expected diffusion as predicted by these closures are identical and are the same as that of the least As we shall mean square estimation (LMSE) m0de1.~~~~’ see,these two distributions are very useful in characterizing the mixing behavior. In general, if the PDF exhibits tails broader than Gaussian, the normalized conditional dissipation portrays a “basin”shaped curve (concave up) near the mean scalar value. For PDFs with tails narrower than Gaussian, E @ ) adopts a “bell”shape (concave down) curve. It is interesting to examine the behavior of the PDF for the AMC. Here, it is more instructive to consider a nonsymmetric PDF within the finite domain -1 = & I 4 I4%= 1, (4) # 0. The PDF in this case is similar to that given by eq 2, mapped into the domain 4 E [-1,13. (30)Sinai, Y. G.;Yakhot,V.Phys.Reu.Lett. 1989,63 (18),1962-1964. (31)Yakhot, V . Phys. Reu. Lett. 1989, 63 (18),1966-1967. (32)Goldhirsch, I.; Yakhot, A. Phys. Fluids A 1990,2 (8),1303-1305. (33)Yakhot, V.;Orszag, S. A.; Balachandar, S.;Jackson, E.; Sirovich, L. J. Sci. Comput. 1990,5 (3),199-221. (34)Jayesh, Warhaft, 2. Phys. Reu. Lett. 1991, 67 (26),3503-3606. (35)Jayesh; Warhaft, Z.Phys. Fluids A 1992, 4 (lo), 2292-2307. (36)Dopazo, C.;OBrien, E. E. Combwt. Sci. Technol. 1976,13,99112. (37)OBrien,E. E. In ~ r b u l e nReacting t Flours;Libby,P.A., Williams, F. A., Eds.; Springer-Verlag: Heidelberg, 1980; chapter 6, pp 185-218.
830 Energy & Fuels, Vol. 7, No. 6,1993 Table 1. Conditions for LEM Simulations case
(6)
LQ/L
0.5
1 2
case 3 4
'Iz 1
0.5
LQIL 2 'I2
5.0
(6) 0.5
0.75
4.0
Substituting this equation into eqs 18 and 19, after significant manipulations, it can be shown that24
E(4,t)=
dG exp(-2(erf 1(4))2)
?rG(1+ G2) dt
(25)
erf1(4)] ex~((-erf'(4))~) (26) o,o
For the nonsymmetric PDF it is not possible to derive a closed-form equation for the variance. Therefore, G cannot be represented in terms of E. Nevertheless, the results reveal some important properties of the AMC. Equation 25 indicates that E(4,t) is always symmetric with respect to 4 = 0 and is independent of the magnitude of the mean scalar value ((4)). This is surmised to be nonphysical. The dependence on the mean value is exhibited through the conditional expected diffusion (eq 26). The results for the conditional dissipation portray a self-similar form in that the dependence on 4 is separable from the dependenceon t (through G). This self similarity is not the case for the conditional diffusion unless (4) = 0. For a symmetric PDF (( 4) = 01,the equations can be simplified further. In this case, it is easy to evaluate E through eq 17. With the use of eq 10, we have
(28) These results are very useful for the analyses of the data obtained from LEM. 4. Linear Eddy Modeling Details of the LEM are described by Kerstein.14 The most important feature of the model in applications to turbulence simulations is its capability to explicitly differentiate among the different physical processes of turbulent stirring (convection) and molecular diffusion (and chemicalreaction in a reacting flow). This is achieved by a reduced one-dimensional description of the scalar field which allows the resolution of all length scales even for flows with relatively large Reynolds numbers. The physical interpretation of the one-dimensional domain is dependent on the particular case under consideration.161e Along the one-dimensional domain, the diffusion process is implemented deterministically by the solution of the appropriate diffusion equation. The manner by which
.
Frankel et al.
\
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0.2
x.
-
,*--..,-a
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Figure 1. PDF of the variable 4 at several u values (case 1).
turbulent convection is treated constitutes the primary feature of LEM. This process is modeled by random rearrangement (stirring) events of the scalar field along the domain. The rules by which these rearrangement processes occur are established in such a way that the random displacementsof fluid elements mimic the effects of turbulent diffusivity. The parameters which govern this process are A, the frequencyof stirring, andf(C), which is a PDF describing the distribution of the segment sizes, C (eddysize)which are to be rearranged. In the application of the model to high Reynolds number turbulent flows, Kerstein2 derives:
Here, TL is the eddy turnover time, L is the integral scale of turbulence, and 9 is the Kolmogorov length scale. Equations 29 and 30 are based on scaling relations for high Reynolds number and therefore, the results are most applicable under this condition. As indicated by and McMurty et ~ 1 caution . ~ must be Cremer et exercised in interpreting LEM results for low Reynolds number mixing. Cremer et ~ 1 . have 3 ~ shown that results from AMC and moderate Reynolds number DNS can be reproduced in remarkabledetail by using a reduced length scale distribution in the model. In particular, a single length scale formation provides very good predictions of the DNS data.41 The length scale distribution in this case is simply replaced by
f(e) = q c - L*) (31) This has been interpreted by Cremer et to reflect the fact that moderate Reynolds number DNS does not necessarily represent the wide-banded stirring process characteristic of a Kolmogorov spectrum. Here, results are compared only to the Kolmogorov length scale distribution as these results should be more applicable to (38) Tennekes, H.;Lumley, J. L. A First Course in Turbulence;MI" P r w : Cambridge, MA, 1972. (39)Cremer, M. A.; McMurtry,P. A.: Keretein, A. R. P h w . Fluids A.. 1993, submitted for publication: (40)McMurtry,P. A.; Ganeauge, T.C.; Kerstein, A. R.; Krueger,S. K. Phys. Fluads A 1993,5 (4), 1023-1034. (41) Eawman, V.; Pope, S . B.Phys. Fluids 1988,31 (3), 506-620.
Binary Scalar Mixing in Homogeneous Turbulence
Energy & Fuels, Vol. 7, No. 6, 1993 831
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flows of practical application. The differences between AMC, DNS, and LEM under this condition are of interest. 5. Results
Several simulations are conducted by means of LEM. The procedure is similar to that of McMurty e t aL40 In accord with the philosophy of LEM, the scalar field is initialized in one dimension with alternating slabs of 4 = 0,l. This implies an initial double 6 distribution for P(4,O). Two different magnitudes of the mean scalar value ( 4 ) p (J)are considered (4) = 0.5, and (4) = 0.75. The former
corresponds to a stoichiometric mixture and the latter to a fuel-rich mixture. With this initialization several different cases are investigated. Table I provides a listing of the parameters employed. Different ratios of L+/Lare considered, were Lb denotes the integral length scale of the scalar at the initial time. The magnitude of the Reynolds number based on the integral scale is set to ReL = 90, and the molecular Schmidt number is taken as unity in all simulations. Results for the symmetric PDFs are shown in Figures 1 and 2. The general trend is somewhat similar in all cases in that the PDF evolves from an initial binary state to an asymptotic distribution centered around the mean value (shown in Figure 1 only for case 1). The difference between the cases is most pronounced at the intermediate and the final stages of mixing. The form of the PDF at a rms of u = 0.13 is shown in Figure 2, where the results are also compared with the Gaussian, Laplace, and AMCgenerated distributions. Each PDF is parameterized with the same magnitude of the variance. This figure indicates that the initial forms of the PDF as generated by AMC and LEM show better agreement for higher values of L+/ L. This was observed by McMurtry et ~ 1 . who ~ 0 indicated that for large Lb/L the small-scale eddies play a less significant role in the overall mixing process. The results of the normalized conditional expected dissipation of the scalar are presented in Figures 3 and 4. The behavior at the initial stages of mixing is similar in all cases, and the curves are bell shaped (shown only for case 1 in Figure 3). However, the difference between all the cases is revealed at later stages of mixing. The results for u = 0.13 are compared with those corresponding to a Laplace PDF and the AMC-generated solution. These are given by eqs 23 and 27, respectively. Note that the normalized conditional dissipation corresponding to the Gaussian distribution is a straight line at unity. The LEMgenerated results of the conditional expected dissipation show that both bell- and basin-shaped distributions are obtained. For cases with PDFs close to the Laplace distribution, the results are close to eq 23 near the mean value. For those cases in which the PDFs are close to those of AMC, the curves show a bell-shaped distribution in agreement with eq 27. Also, consistentwith the analyses in refs 30 and 33 if the tails of the PDFs are broader than the Gaussian (Figure 2a) the conditional expected dissipation curve is basin shaped (Figure 4a). If the skirt of
832 Energy & Fuels, Vol. 7, No. 6, 1993
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the PDF is less extended than that of the Gaussian (Figure 2c),the conditional dissipation is more bell shaped (Figure 4c). The profiles of the PDFs and the conditional expected dissipation for the nonsymmetric initial PDF (case 4) are shown in Figures 5 and 6. These figures indicate that at the initial stages of mixing the behavior of the conditional expected dissipation is similar to that of the symmetric case. However, at later times it looses its symmetry and the rate of boundary encroachment is not the same at the two bounds. The same behavior is also observed in the DNS, as the data of Madnia et a1.6 are analyzed and presented here in Figure 7. The results in this figure are
4
Figure 7. Conditional expected scalar dissipation at several u values (extracted from DNS result&.
also compared with the solution of the AMC. The drawback of AMC, portrayed in this figure, is due to the inability of the model to account for the migration of scalar bounds as mixing proceeds.ls This is demonstrated further by considering the behavior of the conditional expected diffusion, shown in Figure 8 for the symmetric (case 1of LEM) and nonsymmetric (case4 of LEM and DNSS) PDFs. Consistent with the predictions of AMC and LMSE, at final stages of mixing the conditional expected diffusion
Binary Scalar Mixing in Homogeneous Turbulence
Energy & Fuels, Vol. 7, No. 6,1993 833 1 . O k
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calculated by LEM and DNS approaches the mean value of the scalar. However, at regions away from this mean there is a substantial difference between the LEM and DNS results and those of the closures. Note that LEM predicts the trend shown by DNS in that as mixing proceeds, the scalar bounds move inward. This behavior cannot be portrayed by any of the closures, and the lack of agreement between these closures and LEM results becomes more pronounced at later times. Despite the differences between AMC and LEM in predicting the conditional statistics, it is of interest to evaluate the performance of the analytical results based on the AMC for predicting the mean rate of reactant conversion (section 2). This assessment is important from an engineering standpoint, since the mean values and not the details of the conditional statistics are of primary interest. The normalized values of mean oxidizer mass fraction for both the symmetric initial PDF (stoichiometric mixture) and the nonsymmetric PDF (fuel-rich mixture) are presented in Figures 9 and 10,respectively. The results are presented in terms of the intensity of segregation, I,, for all the results based on LEM. These figures suggest good agreement between AMC and the LEM data for all cases. For the stoichiometric mixture the results based on Toor's hypothesis11 and the 3E closurez8are also in good agreement with LEM results, but are not as good as AMC. In the fuel-rich mixture, only the AMC is considered as the other two closures cannot be implemented. In this case also in spite of the significant differences between
834 Energy & Fuels, Vol. 7, No. 6,1993
the AMC and LEM in predicting the conditional statistics, the mean rate of reactant conversion is well predicted. In interpreting these observations, it is important to keep in mind the shortcomings associated with each of the techniques involved (DNS, LEM, AMC). LEM is most applicable to high Reynolds numbers flows, since the scaling laws used in formulating the model parameters assume just that. It is a mechanisticallysound formulation which explicitly incorporates the physics of both molecular diffusion and turbulent stirring. However, artifacts can be introduced due to the one-dimensional formulation and the instantaneous rearrangement event process, particularly in the early stages of ev~lution.~ DNS provides an exact description of the mixing process but is limited to moderate Reynolds numbers in simplified geometries as a result of the extensive computer requirements. AMC shows excellent agreement with DNS in moderate Reynolds number applications, but its lack of explicit incorporation of turbulence properties or molecular properties (Le.,Reynolds number and Schmidt number effects) leaves questions remaining regarding its application in more general mixing configurations. Collectively, the results
Frankel et al.
presented here illustrate the complex and subtle nature of the mixing process and the need for continued research in this area. In the meantime,the analytical results derived in refs 5 and 9 are shown to provide an excellent means of predicting the limiting rate of mean reactant conversion in homogeneous turbulent flows with nonpremixed reactants.
Acknowledgment. We are indebted to Dr. Alan Kerstein for introducing us to the concept of LEM and for many useful technical discussions. We are also grateful to Professor L. Douglas Smoot for inviting us to present this paper at the Seventh Annual Technical Conference of ACERC. This work is part of a research effort sponsored by the National Science Foundation under Grant CTS9012832. Acknowledgement is also made to the Donors of the Petroleum Research Funds administrated by the American Chemical Society for the support of this work under Grant ACS-PRF #25129-AC7E,6. Computational resources are provided by NSF through NCSA at the University of Illinois.
