3640
Znd. Eng. Chem. Res. 1995,34, 3640-3652
Evaluation of Closure Models for Turbulent Reacting Flows Andy D. Leonard,? R Cushing Hamlen,’ Robert M. Kerr,P and James C. Hill* Department of Chemical Engineering, Iowa State University, Ames, Iowa 5001 1-2230
Single-point closure models for single-step chemical reactions in decaying homogeneous turbulence-and some associated experiments and simulations-are briefly reviewed. Three of these models for the reaction term in the concentration field equations are evaluated by comparison with direct numerical simulations. In agreement with other studies of non-premixed single-step reactions, the reactant concentration variance shows little sensitivity to Damkohler number, although the segregation coefficient depends strongly on it. Toor’s model, which hypothesizes that the reactant concentration covariance is independent of the Damkohler number, agrees with simulation data better than do the models of Patterson and of Dutta and Tarbell; however, a n attempt to remove a shortcoming of Toor’s hypothesis by using a n initially Gaussian conserved scalar did not improve its behavior.
Introduction Turbulent flows in the laboratory cannot be predicted reproducibly or in detail, because uncertainties that are present in initial and boundary conditions are rapidly amplified, yet average quantities can often be measured and are reproducible. Even for reacting systems, the desired description of the flow is usually in terms either of a few statistical moments, such as the mean and root mean square values of such quantities as velocity, temperature, and concentration, or of the probability distributions for these quantities. In this paper we (1)briefly review some simple, singlepoint closure theories or models, previous simulations, and pertinent experiments for the case of homogeneous turbulent mixing with single-step chemical reactions, and then (2) evaluate predictions of three of these theories that deal with the reaction rate or source term in the statistical moment formulation of the problem. This is done by comparing the predictions of the models t o data from direct numerical simulations (DNS’s). In DNS, the unsteady, three-dimensional differential equations governing the flow are solved without modeling. Preliminary results of tests of reaction rate closure models with DNS data have been reported by Leonard and Hill (1986, 1988, 1989); a similar study was made by McMurtry and Givi (19891, who used one of the models examined here as well as some probabilistic models examined in Leonard and Hill (1991).
sively by Hill (1976), Jones and Whitelaw (19821, O’Brien (1986),and Borghi (1988). Methods of carrying out direct simulations of reacting flows have been reviewed by Oran and Boris (19871, Jou and Riley (1987), and Givi (1989). Statistical methods are divided into two categories here: probability and moment methods. Probability methods are those which use equations for probability density functions (pdf s) that are generated from the conservation equations governing the system, while moment methods involve various kinds of approximations of the statistical moments, although they may use an assumed shape of a pdf as part of a model. Only single-point methods are considered here. Probability Methods. Joint single-point probability density functions (pdf s) of the reactant concentrations, shown in Figure 1for example, contain all the information needed to calculate single-point statistics of the concentration fields, including the mean reaction rate. Dynamical equations for these pdfs can be derived directly from the governing differential equations for the system (Hill, 1970; Pope, 1985). For example, the joint pdf of the concentration of reactants A and B is the solution to an equation of the form
+ transport terms + transient terms
Background Results from numerical simulations are used in this paper to evaluate simple statistical methods or closure theories for the mean rate of reaction in chemically reacting turbulent flows. In this section, as mentioned above, we briefly review these models, supporting experimental work, and previous numerical simulations which have been used in attempts to validate some of the models. Statistical treatments of turbulent reacting flows have been reviewed previously and more exten-
* Author to whom correspondence should be addressed. E-mail:
[email protected]. FAX: 515-294-2689. Present address: CFD Research Corp., 3325-D Triana Blvd., Huntsville, AL 35805. 1: Present address: Medtronic Promeon, 6700 Shingle Creek Pkwy., Brooklyn Center, MN 55430. Present address: National Center for Atmospheric Research, P.O. Box 3000,Boulder, CO 80307. +
*
I
where f.da,B)is the joint pdf, the top term on the right hand side is part of the flux of probability due to the chemical reaction (where Sa = - k ~ @ is the functional form of the production rate of species A-actually disappearance in the case of irreversible second order reaction A B products), the second term is part of the flux of probability due to mixing by molecular diffusion, the third term arises because of gradient transport, and the last arises because of nonstationary turbulence. For steady, homogeneous turbulence the last two terms drop out. The reaction term is closed, leaving only the second term involving conditional expectation of @A, (D@alA,B),to be modeled. However, this term is known to depend on reaction rate, even though the first term is closed (Leonard and Hill 1991). A further difficulty is that the pdf problem has higher
+
-
0888-588519512634-3640$09.00/0 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3641 dimensionality than the comparable moment closure problem described below, and so the computational cost is high. Nevertheless, pdf methods are being developed because of the potential for dealing with higher degrees of nonlinearity than can be dealt with in the moment formulation. The earliest methods include the LMSE (linear mean square estimation) closure (Dopazo and O'Brien, 1974) and the C/D (coalescencddispersion) method; the latter requires stochastic models to solve; both approaches have limitations (KosAly and Givi, 1987;Leonard & Hill, 1991). More recent approaches being tried include the mapping closure of Kraichnan (Chen et al., 1989; Pope, 1991; Frankel et al., 1992, 1993a; Madnia et al., 1992), the LEM (linear-eddy model) of Kerstein (1992;see also Frankel et al., 19951, and a FP (Fokker-Planck) stochastic model-which includes both concentration and concentration gradient-by Fox et al. (1992). All of these show some promise but are much more computationally demanding than the moment methods. Moment Methods. The equations governing the transport of mass and momentum are typically averaged using Reynolds averaging or, in the case of variable density flows, Favre, or density weighted, averaging (Favre, 1965). Single-point closure models for these equations vary in the level of sophistication but have been well developed for nonreacting flows (Reynolds, 1976;Reynolds and Cebeci, 1978;Launder, 1978;Lumley, 1980). Reacting flows, on the other hand, present certain unique problems when treated with moment methods: concentration values are bounded and positive, pdf s of the concentrations are therefore generally far from Gaussian, and correlations between species concentrations and exponential functions of temperature are possible for nonisothermal cases because of Arrhenius forms for the reaction rate. The Reynolds averaged form of the governing equation for species A in homogeneous turbulence is
where A and B , the concentrations of species A and B, are decomposed into means, and E, and deviations, a and b, from the means. Transport terms do not appear because the fields are homogeneous. Equation 2 and a corresponding equation for the mean value of species B contain a covariance of the concentrations of the two reactants. Higher order moment equations can be derived from the governing equations using averaging rules. For example, equations for the variance and covariance of the reactant concentrations are
The closure problem for reacting flows is similar to that of ordinary turbulence theory: even with the simplifications of statistical homogeneity, the number of statistical quantities is greater than the number of equations. Relationships between the statistical quantities, in addition to the moment equations derived from the governing equations, are needed to obtain a closed system of equations. Only the terms that arise in the moment equations because of the source terms in the governing equations are considered in this paper. Terms involving molecular mixing are discussed by Leonard and Hill (1991). One way t o model the mean reaction rate and other single-point statistics of the concentration fields is to model the joint pdf of the concentrations of the reacting species instead of solving a differential equation, in order to calculate the average reaction rate in a moment method. This type of closure is referred to as an "assumed-pdf" model (Borghi, 1988). For the simple twospecies reaction that is being considered here, either the joint pdf of A and B or the joint pdf of A and the conserved scalar variable (A - B ) can be used to calculate the mean reaction rate. The latter may be preferable, since the marginal distribution of (A - B ) will be independent of reaction rate if the diffusivities of the two species are equal. The assumed-pdf approach is often used in the limiting case of very fast chemistry, since only a onedimensional pdf for the conserved scalar is needed for a single-step reaction. Suggested distributions for the conserved scalar pdf include Gaussian, clipped Gaussian, beta, uniform, bimodal, etc. (Jones and Whitelaw, 1982;Borghi, 1988). Moment Methods: Toor's Model. One of the older models that has received a great deal of attention is one proposed by Toor (19691,which uses the limiting cases of very fast and very slow reactions t o form a closure for intermediate rates. Toor considered a multijet reactor with two feed streams, whereby each stream is introduced in a fraction of the inlets with fixed concentrations of the two species, A and B. The reactants A and B aFe assumed t o have identical mass diffisivities. The conservation equations and the initial conditions for the two species are made identical with a simple change of variables. These equations are identical to the conservation equation for an inert species as the reaction rate approaches zero. The variance of an inert species, the variance of reacting species, and the covariance of reacting species, therefore, all decay at the same rate if the reaction rate is very slow and if the diffusivities of the reactants are equal. Defining the normalized variance-of the concentration of a n inert species to be om2E d2/r$20and the normalized --covariante of reactant concentrations to be q2 = ab/ab,, then q12
and
- a2b
+ ab2) (4)
where we have assumed that the diffisivity D is the same for the two species. The equation for the variance of species B is given from eq 3 by interchanging the variables for the concentrations of species A and B.
= a, 2
The zero subscript denotes an initial value, and the subscript m refers t o a process with mixing in the absence of reaction. In the fast chemistry limit Toor (1962)used a Gaussian pdf for the conserved scalar variable (A - B ) t o relate the mean concentration of each reactant for a stoichiometric mixture to the variance of the value of the conserved scalar, obtaining
3642
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
The covariance can be expressed in terms of the mean concentrations if the - reactants are completely segregated, ab = -A B , and so the covariance is again related t o the conserved scalar variance 2
= om
(7)
Since the expressions are the same for the very fast and very slow cases, Toor (1969)proposed that the relationship may hold for intermediate cases and, therefore, that reactant concentration covariance is independent of the reaction rate. This assumption is commonly referred to as Toor’s hypothesis, or Toor’s model. The initial conditions for most of the cases considered in this paper-complete initial segregation of reactants A and B-have discrete initial joint pdfs for A and B, with only two possible events. The pdf of the conserved scalar is bimodal in this case. Kosaly (1987)has shown that the asymptotic behavior of Toor’s model for the reactant covariance can be modified for the case of very fast chemistry where the initial pdf for the conserved scalar is not Gaussian. (This dependence was noted earlier and parenthetically in a paper by Miyawaki et al. (19741.1 If it is assumed that the pdf approaches a Gaussian shape at long times, as shown from DNS’s by Eswaran and Pope (1988),a simple multiplicative factor that depends on the shape of the initial pdf (in this case, 21n) is needed to correct Toor’s theory for long times, but values at intermediate times are not predicted. No predictions can be made about the behavior of finite rate chemistry, although Kosaly assumes that the fast and slow cases are limits between which the intermediate results should lie. McMurtry and Givi (1989) have shown from DNS that the ratio of the reactant covariance to the conserved scalar variance is between 1 and 2/n for finite-rate chemical reactions, as well as for the limiting case of infinitely fast reaction rate. The fast chemistry limit of Toor’s theory holds when the shape of the standardized pdf, f ’ = afz), where z = (I+ - p)/u and p and u are, respectively, the mean and standard deviation, is constant (Givi and McMurtry, 1988). To illustrate the behavior of Toor’s model for a case where the conserved scalar pdf has a form that can be expressed in terms of the first two moments and that changes shape, assume the pdf of the conserved scalar is a beta distribution (Leonard, 1989; Madnia et al., 1992; Frankel et al., 1993a1,
for I+ defined on the range (O,l), so the conserved scalar must be scaled accordingly. The mean value of the conserved scalar, given by this pdf, is pl(p q ) and the variance is pql(p qI2(p q 11,where the ratio p/q is constant since the mean value of the conserved scalar does not change. The parameters of the beta distribution can be expressed in terms of the mean and variance of the conserved scalar concentration. If the initial concentrations of the reactants are in stoichiometric proportion, then
+
+
+ +
1
1
p=q=--8om2 2
The decay rate of the conserved scalar, therefore, determines the parameters p and q and the shape of the pdf. As p and q go to zero, the pdf approaches two
00
Figure 1. Joint pdf of the reactant concentrations in run A at time t = 3. The origin is in the right corner, and the initial distribution was bimodal with spikes at the top and bottom corners (Reprinted with permission from Leonard and Hill, (1991). Copyright 1991 American Institute of Physics.)
0.9
0.8 0.7
0
1
ot
3
4
5
Figure 2. Predictions of Toor’s theory, assuming a beta distribution for the conserved scalar.
delta functions, a t I+ = 0 and )I = 1, and a s p and q become infinitely large, the pdf approaches a Gaussian shape. Toor’s theory can, therefore, be used to evaluate the mean reactant concentrations and the reactant covariance for the fast chemistry limit in terms of the incomplete beta function, as also noted by Madnia et al. (1992) and Frankel et al. (1993a). The ratio of the reactant covariance to the conserved scalar variance is plotted in Figure 2 as a function of ut, where the conserved scalar variance is given by a, = e-wt. The asymptotic limit of 21n is approached only after the variance has decayed to a value that results in greater than 90% completion of the reaction. While Kosaly’s remarks about Toor’s theory are correct, it is difficult to incorporate them into a closure model since they hold only for the asymptotic behavior of the infinite rate limit. Moment Methods: Typical Eddy Models. Pdfs can be continuous or discrete functions, or a combination of both. A discrete form of the pdf implies that the domain can be divided into a number of representative samples or “typical eddies”, as proposed by Donaldson (1975) and Donaldson and Varma (1976). There are three variables associated with each of these “eddies” or “environments”, in the terminology of Tarbell’s mechanistic closures (Tarbell and Mehta, 19861, that are possible for a joint pdf of two reactants. These are the concentrations of each reactant in the representative sample and the probability of the event itself. The representative samples will be referred to as environments, rather than typical eddies, in this section. The joint pdf of species A and B can be written for n environments as
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3643 Table 1. Parameters for Patterson’s Model environment CI ai
2
A’-+ a’
A
b2
0
-
E2+ b2
Table 2. Parameters for Tarbell’s Model
Si E’ + b2 B
where €& and bj are the concentrations in each of the environments. If a low-level closure is desired, then the number of events must be kept small, and some mechanisms must be proposed to relate some of the variables. One example of a discrete pdf has been proposed by Patterson (1981). Three environments are possible. In the first environment only A is present, in the second only B, and in the third both A and B are present. This model has nine variables, but the probabilities of the environments must add to 1, and the concentration of one reactant has been specified in two of the environments. The model further stipulates that the concentration of A or B in the environment where both are present is the same as in the environment where it is present alone. The parameters in eq 10 for this model are summarized in Table 1. This pdf will result from one-dimensional top-hat concentration profiles for each species that overlap. With these restrictions, the model has four parameters. The mean and variance of the model can be calculated in terms of these parameters and the system inverted to give the parameters in terms of the means and variances of the concentrations of the two reactants. Patterson’s closure requires the solution of eqs 2 and 3 for the mean and variance of the concentration of each reactant. The term ab and the third-second-order order terms a2b and ab2 are modeled with the pdf. The unknown moments of reactant concentration on the right hand sides of eqs 2 and 3 can be expressed in terms of the moments on the left hand side, giving the reactant concentration covariance as
environment 1 2 3
Ci
142 142 1-I,
ai G O 0
A
f%
0
B O
B
taining only one species remain the same, and the probabilities of the occurrence of the environment (or the volume fraction of the environment) decay exponentially, with a characteristic time determined by the mixing of a n inert species 4:
(13) The characteristic time, z, is the only parameter in the model. The probability of occurrence of the third environment is fixed by the requirement that the total probability be unity. The concentrations in the third environment are assumed to be the mean concentrations of the reacting species. The parameters appearing in eq 10 for Tarbell’s model are summarized in Table 2. Tarbell’s closure requires the solution of eq 2 for the mean concentration of each species. A time scale for mixing, or the microscale for the inert species concentration, must be supplied for this model. This time scale can also be used to close the dissipation term. The unknown term in eq 2 can be written in terms of the mean reactant concentrations and a characteristic time as
We have used a slightly modified version of Tarbell’s model since the mean concentration, when evaluated from the assumed form of the joint pdf, does not change. This does not mean that the rate of change is zero when eq 14 is used to close the moment equations, but this form is not really consistent with the three-environment model. We use, instead, I
--
-ab= -a 2 b2IAB
(11)
Patterson recommends using a2b = 0 and ab2 = 0 instead of the formed predicted with the pdf,
2 -a2b2( ab-_-
a2\ 1-=
(12)
in order to improve the prediction of the model. The dissipation term must still be modeled, as the pdf form makes no assumptions about concentration gradients. Patterson recommends using Corrsin’s (1964)form for the microscale for the dissipation rate of concentration fluctuations of a nonreacting species in a n isotropic mixer. Another discrete pdf has been proposed by Dutta and Tarbell (1989)and was derived from a mechanistic model of mixing (Tarbell and Mehta, 1986). The form is similar to Patterson’s in that there are three “environments”. Two environments contain only species A or B and the third contains both. In Tarbell’s model, however, the concentrations in the environments con-
which is obtained by requiring the assumed form for the pdf to predict the correct mean reactant concentration. Other Models. Because concentration values must be positive, concentration -moments are subject to various inequalities, e.g., ab L -AB. Lin and O’Brien (1972)have proposed a model based on such inequalities for third-order moments, but find that different choices for the coefficients can give results differing from each other by an order of magnitude and, in some cases, with opposite signs (O’Brien, 1986). Another interesting idea has been to combine two types of closure models within the same application, although this often involves multipoint methods (e.g., mapping and EDQNM by Frankel et al. (1992);EDQNM, LMSE, and correlation function similarity by Tsai and O’Brien (1993));this method has shown some promise, although there have been very few attempts at it. One of the most interesting of the newer techniques is the so-called conditional moment closure or CMC (Bilger, 1993;Me11 et al., 1993, 1994);the premise is that equations for such variables as reactant concentration, conditioned on levels of a nonreacting conserved
3644 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 Table 3. Various Experiments on Mixing with Chemical Reaction Damkahler no. work reaction (A B) DalI = k&AgalD Vassilatos and Toor NaOH HC1 10’ (1965) LiOH HC1 2LiOH HOOCCOOH LiOH HCOOH 1 COZt 2NaOH CO, “3 10-2 HCOOCH3 NaOH 10-3 107 Mao and Toor (1970) HC1+ NaOH maleic acid OH104 103 nitrotriacetic acid OHCOZ 2NaOH 1 0.1 Bennani et al. (1985) CHzCOOCzHs NaOH HCOOCH3 NaOH 40 Mudford and Bilger 0 3 NO3 1 (1984) qjmera et al. (1976) 0 3 NO3 1
+
+
+ + +
+
+ +
+
+ +
+
+
+
scalar, are more readily and accurately closed or modeled than for the nonconditioned variables. Such closures as these will be addressed in future papers. Experimental Studies. Several experimental studies have been performed with simple second-order reactions in decaying, nearly homogeneous turbulence, primarily to investigate features of Toor‘s theories (1962, 1969). Thse studies, spanning a large range of reaction rates (lolo),are listed in Table 3. Vassilatos and Toor (1965) studied reactions with a wide range of kinetic time scales in a tubular multijet reactor. The rates for acid-base neutralization reactions were controlled by mixing, whereas the rate of hydrolysis of methyl forgate was controlled by the kinetics. The time scales for mixing and reaction were comparable for an intermediate case of C02 NaOH. The results of the experiments with the very rapid reactions supported Toor’s 1962 theory relating mixing to conversion. Mao and Toor (1970) studied acid-base reactions between the moderate (Da= O(1)) and very fast (Da= 0(107))reaction speeds of Vassilatos and Toor’s earlier work. The reactor was the same as that used by Vassilatos and Toor, but the mixing device was improved by adding more tubes to feed the reactant streams. In both of these studies small temperature rises were measured and related to the conversion of reactant species. Turbulence statistics were not measured. The invariant hypothesis of Toor (1969) was tested by using the fastest reaction to calculate the mixing characteristics of the reactor. The experimental data supported Toor’s predictions, although there was enough scatter in the data to also support the correction, made by Kosdy (1987), to Toor’s theory. McKelvey et al. (1975) studied the velocity fields and mixing in a model of the reactor used by Vassilatos and Toor (1965) and Mao and Toor (1970). Complicated flow patterns near the inlet of the reactor were found, indicating the flow was not truly homogeneous. The variance of an inert species decayed at t-=, as predicted by Hinze (1975) for initial period turbulence. A more stringent test of Toor’s hypothesis was made by using the measured velocity and mixing data to integrate the mean concentration equations. The results showed that Toor’s model could be used to predict the mean concentration for finite reaction rates, in support of his 1969 theory. Bennani et al. (1985) studied the very slow alkaline hydrolysis of ethyl acetate and the fast hydrolysis of
+
methyl formate, using grid-generated turbulence with injectors for one of the reactants located in the grid. The methyl formate reaction was studied by both Bennani et al. and Vassilatos and Toor (1965). This reaction has different speeds for the two studies, relative to the mixing rate, because the reactor used by Vassilatos and Toor was more efficient in mixing the reactants. Conductivity probes were used to infer the concentrations of the reactants. Power law decays of the turbulence intensity and of the concentration variance of an inert species were in agreement with previous studies (Sreenivasan et al., 1980). The data for the fast reaction case supported Toor’s (1969) hypothesis. Ajmera et al. (1976) and Mudford and Bilger (1984) have studied the gas-phase reaction between ozone and nitrous oxide. Mudford and Bilger used a turbulent smog chamber for the nitrous oxide-ozone reaction. Opposingjets containing the reactants entered one end of a long cylindrical bag. Ajmera et al. used an annular reactor for this reaction that was designed to provide less rapid mixing than the reactors that were used by Vassilatos and Toor (1965) and by Mao and Toor (1970). Ajmera et al. concluded that Toor’s model applies t o gases as well as liquids, and that the mixing data from very fast liquid reactions can be used in the design of gas-phase reactors. Mudford and Bilger used the results of their experiments t o test closure theories for the mean reaction rate. Joint pdfs of the composition were obtained from chemiluminescent analysis. The model that gave the best agreement with their data was the one based on perturbations from the fast chemistry limit (Bilger, 1980). Although many of the experiments showed general support of Toor’s (1969) invariant hypothesis, the issue raised by Kosaly (1987) has not been settled by laboratory experiments. Also, not all quantities needed in the evaluation of Toor’s and other closure models could be measured in these experiments, prompting the use of direct numerical simulations. Numerical Simulations. Direct numerical simulations of a finite rate, bimolecular reaction in isotropic turbulence have been made in a number of studies to test closure models for the mean rate of reaction (Hamlen, 1984; Leonard, 1988; Leonard and Hill, 1986, 1988,1989; McMurtry and Givi, 1989; Gao and O’Brien, 1991). There have also been some simulations for infinite reaction rate to test some theories for the limiting reaction rate (Givi and McMurtry, 1988; Madnia et al., 1992; Frankel et al., 1993a,b), and others t o investigate some kinematical features of the flows (Leonard and Hill, 1990,1992; Nomura and Elgobashi, 1992; Fox et al., 1992). Models proposed by Toor (1969) and Patterson (1981) were tested with the data from the simulations made by Leonard and Hill, which used a lower resolution than in the present study. The form of the reactant covariance, ab, predicted by Toor’s theory was in better agreement with the DNS results than was the form predicted by Patterson’s theory. The predictions for the mean reactant concentration are also better when Toor’s model is used to close the moment equation. McMurtry and Givi used simulations of forced isotropic turbulence t o test Toor’s model at the same resolution as the present study, but larger Reynolds and Damkohler numbers were used. The ratio of the reactant covariance t o the variance of a conserved scalar,
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3646 2
q lam2 =
__
ablab,
421420 and (which is assumed by Toor t o be unity for all reaction rates) approaches 2/nfor the limiting case of very fast chemistry, the value indicated by Miyawaki et al. (1974) and Kosiily (1987) and demonstrated in the infinite Damkohler number simulations and using the beta distribution fit. The deviation of this ratio, which is called the unmixedness ratio by McMurtry and Givi, from unity is small for small values of the Damkohler number and appears t o approach zero for long times. A tendency for the unmixedness ratio to approach unity at long times was seen earlier by Hamlen (1984) and by Leonard (1988) in low resolution simulations. Leonard and Hill (1988, 1989) showed that the microscale for the concentration of reactants was not sensitive to the value of the reaction rate for the case of non-premixed reactants; this was opposite the behavior found by Borghi et al. (1989) for a nonlinear single-species reaction that was intended to emulate a nonisothermal premixed reaction. Simulations have also been made for a two-step reaction (Chakrabarti, 1991; Gao and O'Brien, 1991); the use of simple models to predict concentration covariances and chemical selectivity in these cases is even less successful than in the present study. Description of the Problem The present study was undertaken to reevaluate some of the simple reaction rate closure models discussed above, for the case of a finite rate chemistry, and with higher resolution simulations. We also wanted to see if the behavior noted by Kosaly (1987) for Toor's model could be avoided for suitable initial conditions. The particular case of interest here is the single-step, irreversible, second-order reaction A B products with the rate of reaction given by S = -k*, for the case of stoichiometric non-premixed reactants. The reaction rate coefficient KR is considered to be constant in this study, but in general it will vary with time and position and have a temperature dependence. Statistically homogeneous, decaying turbulence was used in the present study. In this case there are no mean spatial gradients, and so we were able to focus on the chemical reaction term in the statistical equations. Initial turbulence levels are given, instead of considering the problem of how the turbulence was generated. The velocity field was resolved down to the Kolmogorov scale, so no turbulence modeling was used. The velocity field was isochoric, with constant Newtonian viscosity, and the concentration fields were assumed to be passive quantities with respect to the velocity field. The problem is an initial value problem, but is similar t o following the flow downstream in a turbulent, multijet, tubular flow reactor (Hill, 1976). Governing Equations. The governingequations for the problem are the incompressible Navier-Stokes equations and the mass conservation equation for each reactant,
+
-
aA = DV2A - k@ aA + ujq at
(19)
with an equivalent equation for the concentration of reactant B. In eq 17 p is the modified or equivalent pressure, corrected for hydrostatic head (Bird et al., 1960). Fickian pseudobinary diffusion is assumed, with the same constant effective mass diffisivity D for each reactant. Because the reaction rate coefficient is constant, independent of temperature, the energy equation is not needed. Numerical Method. A pseudospectral method was used to integrate the governing equations. (See Kerr (1985) and Hamlen (1984) for more details about the computer code.) In this method the variables are expressed as truncated Fourier series, and the ordinary differential equations for the expansion coefficients are solved. The convolution terms which arise from the quadratic nonlinearities are not evaluated directly; rather, the nonlinear terms are evaluated a t collocation points in physical space, and the Fourier coefficients of these nonlinear terms are then evaluated. A fast Fourier transform (FF")algorithm is used to transform variables between physical and Fourier spaces. The velocities and concentration of species A are expanded as follows: u(x,t) =
v(k,t)enVx
(20)
Ik(cK
A(x,t) =
&k,t)enex lklcK
a
where v and are the Fourier coefficients t o be integrated forward in time. An equation similar to eq 21 is used for B , the concentration of species B. The governing equation for u(x,t) and A(x,t) and the orthogonality properties of e**= can be used to write equations for v(k,t) and &k,t):
and
d&k,t> dt
+ Dk2&k,t) = T(-u.VA
- k@}
(23)
and similarly for species B, where T{...} denotes a Fourier transform. The continuity equation, k*v= 0, has been used to eliminate the pressure term. Equations 22 and 23 were integrated in runs A-E with a compact third-order Runge-Kutta method, using integrating factors derived from the molecular terms, and with a time step determined from a CFL stability criterion (Leonard, 1989). Aliasing errors are reduced by spherical truncation of the Fourier coefficients. The domain for the simulations is a cube of size (W3, with periodic boundary conditions. The calculations were performed with 643 Fourier coefficients on a Cray X-MP computer a t the National Center for Supercomputing Applications (Urbana, IL). Conditions and physical
3646 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 Table 4. Summary of Initial Conditions and Parameters Used in the Simulations
-
--
-
runa
kR
V
3u212
Ri
a2,b2, - ab
A B C D
1.0 1.0 1.0 5.0 5.0 2,8
0.01 0.015 0.02 0.01 0.02 0.02
1.33 1.33 1.33 1.33 1.33 1.55
65.8 43.9 32.9 65.8 32.9 19.7
0.856 0.856 0.856 0.856 0.856 0.748
E Z
I
a All simulations were performed with 643 Fourier coefficients ~ . value of the Schmidt number (Sc on a domain of size ( 2 ~ ) The = v/D)was 0.7, and the initicl mean concentration of each reactant was 1.0 in all runs, except A0 = 1.03 and Bo = 0.97 in run Z.
11.8
11.0
A 11.4
11.1
(1
parameters for the simulations are summarized in Table 4; for runs A-E the Damkohler number of the first kind (Da= k d o h d u ’ ) was equal to k~ in the units of the simulation. A somewhat different procedure was used for run Z, which was carried out on a Cray 2 computer a t the National Aerodynamic Simulation facility at NASA Ames Research Center, and is described more fully by Leonard and Hill (1992). The simulation was made with a pseudospectral code adapted from Rogallo (1981) which uses a predictor-corrector procedure with random grid shifts to reduce aliasing errors. The values of Da used in this simulation were 1.6 and 6.4. Initial Conditions. For computer runs A-E the initial reactant distributions were specified to be slightly damped square waves of one period in the XI direction and uniform in the x2 and x3 directions; this was done for other studies on the effect of scalar anisotropy on the rate of scalar dissipation and on the structure of the reaction zone (Leonard and Hill, 1991,1992). The initial concentration distribution of A was therefore
where kl is an integer and where diffusion has been allowed to act for a small time t* before the simulation to minimize Gibbs’ ringing effects. Species B fills the “slabs” left vacant in eq 24 so that the species are initially segregated, i.e., non-premixed. Volume averages of the concentration gradients are zero, and the reaction zones are well-defined; two reaction zones exist because the boundary conditons for the reactant concentrations are periodic. The initial velocity field is chosen by scaling randomly selected Fourier coefficients to give an energy spectrum of the form
I1
1
4
5
I
Figure 3. Mean concentration of reactant A for runs A-E. The initial Ri is 65.8 for runs A and D, 43.9 for run B, and 32.9 for runs C and E. The initial Damkohler number is 1 for runs A-C and 5 for runs D and E.
generating an isotropic distribution of spatially segregated reactants, fully correlated with the velocity field. Run Z then begins at this condition, with u‘ = 1.03 and Ag = 0.381.
Results and Discussion Results of the above direct simulations were used to test closure models of Toor (1969), Patterson (1981),and Dutta and Tarbell (1989) for the moment equations by (1)comparing the values of the terms to be modeled with the forms predicted by the models, using DNS data for both, and (2) comparing the integrated moment equations for the mean concentration of the reactants for each model-using DNS data in the modeled terms-with the DNA results for the mean concentrations. Since various Reynolds and Damkohler numbers are used in this study, we first summarize the effects of varying these quantities. Reynolds and DamktJhIer Number Effects. The mean concentration of reactant A is shown in Figure 3 for runs A-E. These five runs have three different values for the initial Reynolds number and two M e r e n t values of the Damkohler number. Also shown in Figure 3 is the mean concentration in the limit of infinitely fast reaction, calculated from the conserved scalar (A - B ) in run B. The initial value of 2 in this limiting case is not 1,because the reactants for the finite-rate chemistry cases were not perfectly segregated. A step change occurs in the mean concentrations for the infinitely fast reaction at t = 0, since the reactants cannot coexist. The mean concentration, decreases faster for the two cases with the larger Damkohler number, but not as fast as in the limiting case. The time scales for mixing and reaction are comparable in these cases, and the Damkohler numbers are, thus, moderate to high. It is for these cases, where the mean reactant concentration cannot be predicted from either the kinetics expression or from an inert tracer species (i.e., the conserved scalar), that proper modeling in the closure theories is most important. The mean values of the reactant concentrations are barely changed when the viscosity is changed by a factor of 2. The change in viscosity changes the length scales of the velocity and the scalar fields, rather than affecting directly the rate of decay of kinetic energy and concentration variance. The behavior of the length scales is discussed by Leonard and Hill (1991).
A,
The peak wavenumber Fzo is 2.86, which gives an initial Taylor microscale Ag of 0.70. The initial concentration field for run Z, the isotropic case, was generated by carrying out a presimulation with a conserved scalar C#I having zero mean; the Fourier coefficients for both C#I and the velocity field were selected from Gaussian distributions and had energy spectra that fall off as k-2 and scalar spectra as It-’ a t high wavenumbers; these spectra develop rapidly to an asymptotic shape (Leonard and Hill, 1991, 1992). The presimulation is then stopped, and reactants A and B are assigned nonzero values in spatial regions where C#I takes on positive or negative values, respectively, thus
1
2
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3647 -0.5
-0.7 c
H
x
-0.8
0
1
2
1
3
4
0
5
Figure 4. Development of the variance of the concentration of reactant A for runs A and D.
+
OD
2
1
3
t
run A run B run C runD runE
4
5
-- -
Figure 6. Development of the segregation coefficient, ablA B, for runs A-E.
a
1.2 klllTTllirlrrrrlj
d s dt
-0.3 -0.4 0
1
2
t
3
4
.++ +++
0.6
5
' ' ' " 1 ' ' ' ' 1 ' ' ' ' ' ' ' " ' ' '
0
Figure 5. Development of the contributions to the rate of change of the variance of the concentration of reactant A for runs A and D.
The time evolution of the variance of the concentration of reactant A in runs A and D is shown in Figure 4. The reaction rate coefficient differs - by a factor of 5 for these runs, but the variance, u2, is almost unchanged. There are two terms on the right hand side of eq 3 that contribute t o the rate of change of the reactant concentration variance-scalar dissipation and chemical reaction. The contributions to the rate of change of the variance due t o each of these terms are shown in Figure 5. The largest of the two terms is the scalar dissipation rate, which is a factor of 3-4 greater in magnitude than the reaction term. Even though the magnitudes of both terms increase when the Damkohler number is increased, the signs of the two terms are different, so the net change is small. Consequently, the scalar variance is dominated by molecular dissipation and is not very sensitive to the Damkohler -- - number. The segregation coefficient, ab/A B , is shown in Figure 6 for runs A-E. This parameter is a measure of the difference in the reaction rate from the case with uniform concentrations, as shown by a simple rearrangement of eq 2, (26)
The magnitude of the segregation coefficient increases initially, then decays slowly for cases A-C, with the lower Damkohler number, and remains nearly constant for runs D and E, with the higher Damkohler number. The initial reaction rate is abnormally large, as a result of the smoothing of the initial conditions and the consequent lack of complete segregation, but adjusts rapidly. The rate of change of the mean concentration, compared to the rate for uniform concentration fields, will be less than half the value for the lowest Damkohler number and about 10% for the highest Damkohler number. This illustrates the importance of modeling
b
1
2 , 3
4
5
1.8 1.6 1.4
run E
. 1.2
run 0
1 .o
0.8 0.6 0
1
2 t 3
4
5
Figure 7. hedictions of closure theories for the reactant covariance, normalized with the actual value, for nonisotropic (slab) initial concentration fields: (a) runs A-C (Da = 1);(b) runs D and E (Da = 5).
the source term. Assuming that the mean reaction rate is a function of mean concentrations will most likely produce serious errors. Instantaneous Values of the Modeled Terms. Here we use the DNS data to evaluate the concentration covariances for each model, and compare these model covariances to the actual covariances from the simulations. The predicted forms for the reactant concentration covariance, normalized with the actual values from the simulation, are shown as a function of time for each model for runs A-E (slab initial conditions) in Figure 7. The instantaneous test results for the closure models are also shown for run Z (isotropic initial conditions) in Figures 8 and 9, the latter figure showing the nonnormalized results to give the reader an idea of the absolute errors in the models. The same reaction rate coefficient is used in runs A, B, and C, but the viscosity is different for each run. Runs D and E used the same viscosityas runs A and C, respectively, but a higher value of the reaction rate coefficient was used in the former. The means and variances of the concentration of reactants are insensitive to this change, as are the predictions of the models when data from the simulation are used t o estimate the covariances. The velocity field for run Z is different from runs A-E, as mentioned earlier, but the same trends are seen in the model predictions.
3648 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
t-t
11.1
0 0
c
DNS,runA DNS,runD
1l.IlS
0.5
0
1
3
2
t
II
u
I
2
3
4
5
I
-
Figure 10. Predictions of Patterson's closure theory for the thirdorder moment a2b,compared to the actual value, for runs D and E. 0.0 1
0
t
3
2
Figure 8. Predictions of closure theories for the reactant covariance, normalized with the actual value, for isotropic initial concentration fields (run 2): (a) Da = 1.6;(b) Da = 6.4.
a
0.0
ab
-0.5
-1.0
l-#''X 0
b
" " " " " " " " " ' " " " ~
1
t
2
3
0.0
-0.5
- - - Tarbell 0
1
t
2
3
Figure 9. Predictions of closure theories for the reactant covariance, nonnormalized, for isotropic initial concentration fields (run 2): (a) Da = 1.6;(b) Da = 6.4.
Of the three models, Toor's shows the smallest deviadata. (The quantity plotted for Toor's tion from the DNSmodel is ~closure/abDNS = um2/yj2.)The form of Toor's theory shown here does not account for the changing shape of the conserved scalar pdf. The value of the covariance ratio for an infinitely fast reaction should approach 1.25 (Koshly, 19861, as the trend for the larger Damkohler number data indicates. The predictions of Toor and Tarbell are similar for both values of the Damkohler number, but Patterson's predictions are greater than the simulation results for the higher Damkohler number and lower for the lower Damkohler number. Patterson's model would be expected t o be more accurate for higher Damkohler numbers, since it corresponds to spatially segregated reactants. However,
the model assumes that the concentration of a reactant will have the same value in a reaction zone as in a region containing only that reactant. Characteristic compositions in the reaction zone will actually be much less than the values outside the reaction zone for fast reactions. Neither Tarbell's nor Patterson's model predicts the proper initial value for the covariance, because the reactants are not completely segregated in the simulation, but that may be a defect in our procedure rather than a defect of their models. This lack of initial agreement does not affect the agreement at later times in Figures 7-9, however, since the curves are calculated using the instantaneous statistics. It will, however, affect the initial rate of change of the moments when the statistical equations are integrated, and can, therefore, affect the predictions for moments at all times. Patterson's model also requires a prediction -for the third-order concentration moments a2b and ab2, since the reactant variances are used in the covariance approximation instead of the inert species variance. The deviation from the DNS data of the prediction of Patterson's model for the contribution of reaction to the rate of change of the reactant variance is shown in Figure 10. The simulation results support Patterson's suggestion that the triple moments should beneglected, and also show that the actual values of a2b are of opposite sign to that in the model. Gaussian Initial Conditions. Kosdy's (1986) remarks on Toor's (1969) theory were not taken into account in the preceding tests of the closure, because the rate of change of the conserved scalar pdf from a bimodal t o a Gaussian distribution cannot be predicted a priori. The initial pdf, however, can be defined to be Gaussian in the simulations. Toor's derivation for the covariance in the limit of slow reactions assumed an initially bimodal pdf, and so it is really not valid for such initial conditions. Nevertheless, several 323simulations were performed with initially segregated reactants and a Gaussian pdf for the conserved scalar, as shown in Figure lla,b. To generate this distribution, the conserved scalar (@ = A - B ) was initialized by chosing values from a Gaussian distribution. The initially segregated reactant concentrations were defined by setting the concentration of species B to zero when the conserved scalar was positive and t o the absolute value of the conserved scalar when it was negative. The
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3849
0
A
Da=5 Da=lO
‘ . . ~ ” “ ‘ 1 ” ’ “ “ ’ . ~ . ” .
0
1
2
3
4
5
t Figure 12. Development of Toor’s prediction for the covariance when the initial conserved scalar pdf is Gaussian.
b
0.00 -5.0
-4,O
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
a-P
-60
-60
-4.0
-00
00
20
IO
EO
00
Figure 11. Pdf s for the case of an initially Gaussian conserved scalar for a 323 simulation: (a)joint pdf of the initial concentrations of the reactants (origin a t the lowest corner); (b)initial pdf of the conserved scalar A - B (solid line) compared to a Gaussian (dotted line); (c) development of the standardized pdf of the conserved scalar at times t = 0, 0.2, 0.4,0.6, 0.8, 1.0, and 1.2.
concentration of species A was equal t o the conserved scalar when it was positive and zero when it was negative, Le., (A$) = (&O) if 4 I0 and (O,-#) if < 0. The conserved scalar pdf remained Gaussian during the simulation, as shown for the standardized pdf, f ’( 2 ) = fl(ly-p)/a), in Figure l l c a t various times. The initial variances and length scales are almost, but not exactly, the same for each reactant, because the concentrations are chosen from a random distribution. The initial length scale for the covariance, however, is significantly different than the length scales for the concentration variances, and so the slow-reaction limit of Toor’s theory does not hold for the nonreacting case, as mentioned above. In particular, the slow-reaction limit of Toor’s theory requires the initial conserved scalar pdf to be bimodal. In that case, the s u m of the concentrations of the two reactants remains constant, and the concentration fluctuations of species A and B
remain of equal magnitude and opposite signs, Le., a = 4. The right hand sides of eqs 3 and 4 are, then, identical. When the initial conserved scalar pdf is Gaussian, the relationship between the concentration fluctuations no longer holds. The length scales for the variances and covariances can be different, so the dissipation terms will not be the same. The form of the covariance predicted by TOOF’S theory is shown in Figure 12 for different values of the Damkohler number. The covariance is not independent of the reaction rate for these conditions, and so the agreement of the predictions of the theory and the DNS results for an initially Gaussian conserved scalar are poorer than for any of the cases with the initially bimodal pdf for the conserved scalar. Integral Tests of the Closure Models. The moment equations are now integrated with each model in order to obtain mean concentrations as a function of time and which are compared to the actual values from the DNS. The behavior of the predicted forms of the modeled terms using the DNS data, presented in the previous section, only suggests how well the models will predict the mean concentrations. In order to test these predictions, the ordinary differential equations for the means of variances of the reactant concentrations and the variance of the concentration of an inert species were integrated with the three models discussed above, for each of the DNS studies. The correction factor to Toor’s theory discussed by Kosaly was not included in the calculations because it cannot be incorporated in the closure model, nor was the initially Gaussian case used here. The time scale needed for the closures was supplied by specifying the scalar dissipation microscale as a function of time for each of the runs. With the mixing properly accounted for, the models were only tested on their ability to model the terms due to reaction. The results of runs A and D are selected as representative of the data as a whole, since the value of the viscosity did not affect the agreement between the DNS data and the integrated moment equations. Toor’s model shows excellent agreement with the DNS results for the moderate Damkohler number case (Figure 13a) and reasonably good agreement for the high Damkohler number case (Figure 13b). As described earlier, Toor’s model has considerable experimental support, and it compares with DNS data in this study more favorably than do the models of Tarbell and Patterson. The success of the model for the cases used here may be due, at least in part, to the dominant effect of the initial conditions-especially that of the initial concentration gradients-on the decay of variance and covariance of the scalars. Toor’s model was less successful when a diluent with a nonuniform concentration was present and the conserved scalar pdf was Gaussian
3660 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
0.8 t
1.
t
0.6
- \
will be transported toward the origin. This picture is at odds with the forms assumed by both Tarbell and Patterson. A parametric form that is a continuous function of composition, such as the multivariate beta function proposed by Girimaji (19911, may be required to approximate the pdf.
-
-Toor
--e
Conclusions
0 0
b
1
2
t
3
4
5
1 0.8
A
0.6
0.4 0.2
--
- - h - - Patterson
' * ' ~ " ' ' ~ ~ " ' ' " ' ~ ~ ~ " ' ~
0
3 4 5 t Figure 13. Predictions of closure models and DNS results for the mean concentration of reactant A (a) run A (Da = 1); (b) run D (Da = 5). 0
1
1.o
a2
2
-----
0.8
DNS (Run D)
(Run A) - e - - Patterson Patterson (Run D)
-3-
0.6 0.4
0.2
0.0 0
1
2
3
4
5
t Figure 14. Redictions of Patterson's closure model and DNS results for the variance of the concentration of reactant A from runs A and D.
than when the conserved scalar pdf was bimodal. The contribution of dissipation to the rate of change of the reactant variances and covariances was generally much larger than the contribution from reaction when the initial conserved scalar pdf was bimodal. As for the two typical-eddy closure models, Tarbell's model shows fairly good agreement with the DNS data for run A, but poorer for run D, in agreement with the observation by Gao and O'Brien (1991). Patterson's model does not agree well with the data for either Damkohler number. The shortcomings of Patterson's model can be seen further by comparing the reactant concentration variance, predicted from integrating the model equations (eqs 2,3, and 111, to the DNS results (Figure 14). The modeled form for the rate of change of the variance due to reaction is not accurate, and the variance decay rate is much too rapid. The reason that these models do not perform as well as moderate Damkohler numbers may be that approximating the joint composition pdf as three delta functions is too crude, as suggested by Figure 1,which shows a continuous distribution. The conserved scalar pdf, which is independent of the reaction rate, will presumably approach an asymptotic Gaussian shape. In the fast reaction limit the joint pdf must be zero when both species concentrations are nonzero. All the probability must be spread along the axes in composition space and
Direct numerical simulations of finite-rate chemical reactions in decaying homogeneous turbulence were carried out for moderate and high values of the initial Damkohler number and for moderate Reynolds numbers. The values of the initial Reynolds number had no effect on the development of the mean or variance of the concentration of either reactant or of an inert species. The reactant concentration variance showed very little sensitivity to Damkohler number, since its rate of change was dominated by molecular dissipation. The segregation coefficient, on the other hand, showed a marked change for different Damkohler numbers. At high values of Da, the segregation coefficient maintained a fairly steady value close t o -1. The mean concentration was predicted much better when Toor's model was used to close eq 2 than for the other closure models examined here, although that method has shortcomings that are not improved even for the case of an initial Gaussian form for the conserved scalar pdf. Tarbell's model was able to predict the mean concentration adequately for a moderate value of the Damkohler number, but not for a high value. Patterson's model, the only one that predicted both the mean and variance of the reactant concentrations, was unfortunately the least satisfactory of the three models tested under the conditions of the simulations. The influence of the Damkohler number on the reactant concentration variance was incorrectly predicted, which contributed t o errors in predicting the mean value. One disadvantage of moment methods is that they need to be formulated for a specific case, such as the two-species, second-order reaction studied here. The presumed pdfmethod does have the advantage that any concentration moments can be evaluated, but simple forms are used to restrict the dimensionality and domain of the pdf. The alternative t o using moment methods is t o solve a modeled pdf equation. Moment methods are generally easier to use, but the full pdf method is better suited to the calculation of reacting turbulent flows, since the models for mean reaction rate can be avoided. The pdf equations can be very computationally demanding, on the other hand, because of the higher dimensionally of the problem, generally necessitating the use of a stochastic model as an approximation. One final caveat is needed here, since DNA can be made only for low t o moderate Reynolds numbers, yet in some cases-especially for the stochastic models-the theories are intended to describe mixing with reaction at high Re. This was pointed out by Cremer et al.(1994) in a study of length-scale effects on the LEM model which showed Re dependence of some statistics of the scalar field.
Acknowledgment The authors appreciate the opportunity to contribute to this issue in honor of the classic textbook Transport Phenomena. Computational resources were provided by
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3651 the National Center for Supercomputer Applications, in part under National Science Foundation Grant ECS 8515047,and by the Numerical Aerodynamics Simulation Program at the NASA-Ames Research Center. Support provided by the Phillips Petroleum Co. is also gratefully acknowledged. R.M.K. is with the National Center for Atmospheric Research, which is funded by the National Science Foundation.
Nomenclature A, B = chemical species A, B = concentration of species A or B a = Fourier component of the concentration A(x,t) Ao, - -Bo = initial concentration of species A or B A, B = average concentration of species A or B a, b = fluctuating component of concentration of species A -or B a’ = variance of concentration fluctuations of species A ab = covariance of concentration fluctuations of species A and B DNS = direct numerical simulation D = diffisivity Da = Damkohler number of the first kind (k&,Adu’) DaII = Damkohler number of the second kind (see Table 3) E(k) = initial energy spectrum for the velocity field fiq) = pdf of the variable q f-(a,D) = joint pdf of the concentration A, B of species A and B -Z, = intensity of segregation of inert scalar = qP/#t k = wavenumber vector ko = peak wavenumber of E(k) k R = kinetic rate constant p = pressure Rn = turbulence Reynolds number, u’A& SA= reaction rate of species A Sc = Schmidt number t = time T{...} = Fourier transform u = velocity u’ = turbulence intensity, ui = fluctuating component of the velocity in the i-direction v(k,t)= Fourier component of the velocity field xi = spatial coordinates, i = 1,2,3 Greek Letters a,p = distribution variable for A or B r = gamma function Af = longitudinal integral length scale of the velocity field Ag = Taylor microscale v = kinematic viscosity p = density of the fluid 9 = inert scalar, conserved scalar ~2 = reactant concentration covariance normalized with initial value am2= inert or nonreacting scalar variance normalized with initial value z = characteristic time of scalar variance decay w = decay exponent Symbols = Fourier transformed variable - (overbar) = time-averaged or volume-averaged value
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Abstract published in Advance A C S Abstracts, September 15,1995. @
