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Modeling Turbulent Scalar Mixing as Enhanced DiffusionSharath S. Girimaji aa A. S. & M. inc. , Hampton, Virginia, 23665Published online: 09 Jun 2010.

To cite this article: Sharath S. Girimaji (1994) Modeling Turbulent Scalar Mixing as Enhanced Diffusion, Combustion Scienceand Technology, 97:1-3, 85-98, DOI: 10.1080/00102209408935369

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Combust. Sci. and 1tch., 1994, Vol. 97, pp. 85·98Photocopying permitted by license only

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Modeling Turbulent Scalar Mixing as Enhanced Diffusion

SHARATH S. GIRIMAJI A. S. & M. tnc., Hampton, Virginia 23665

(Received February 7, 1992; in final form January 7, 1993)

Abstract-An enhanced diffusion model is proposed for scalar mixing in turbulence. Using a Lagrangianframe analysis and a model simplification the enhancement in the diffusivity due to the velocity field isestimated. The model explains many of the characteristics of turbulent scalar mixing observed in directnumerical simulations,

1 INTRODUCTION

The effect of turbulent velocity field on scalar mixing has long been modeled usingenhanced diffusivity ideas, e.g., Launder and Spalding (1970) and Bilger (1980). Inthese works, the (mean) turbulent scalar flux is modeled with a gradient-diffusionapproximation where the flux is assumed to be proportional to the mean scalar gradient.The proportionality constant is a so called 'turbulent diffusion coefficient' which isdetermined from the characteristic velocity and length scales of the turbulence. Thesemodels have been used with some success in calculating mean scalar fields in manycombustion applications.

In this paper, a new enhanced diffusion model is proposed for Fickian diffusion inturbulent flows. The basic idea behind the model is the following. Consider two initiallyidentical random scalar fields. One field evolves in the presence of isotropic turbulence;the second field evolves without the velocity field, but its Fickian diffusion coefficientis replaced by a time-dependent enhanced diffusion coefficient. The statement of themodel is that, for the proper choice of the enhanced diffusion coefficient, the scalarprobability density functions (pdt) obtained from the two fields, at any given time, arenearly similar. The enhanced diffusion coefficient is a function of the turbulence fieldparameters.

The present model shares some features with the mapping closure models (Chen,Chen, and Kraichnan 1989; Pope 1991; Girimaji 1992a) but is different in some veryimportant aspects. Much like the mapping closure models, the present model can beused to calculate the evolution of the scalar pdf in homogeneous turbulence. Moreover,unlike the mapping closure models, the enhanced diffusion model (i) offers a descriptionof the evolution of the mean scalar dissipation rate (and hence the time scale of thescalar pdf evolution) in terms of turbulence velocity field properties, and (ii) is likely tobe uniformly valid over all stages of the mixing process. The mapping closure model isknown to be formally invalid during the final stages of mixing (Girimaji 1992a, 1992b).

The enhanced diffusion model is developed in a Lagrangian reference frame in Section2. In Section 3, the model computations are compared against the direct numericalsimulations (DNS) of Eswaran and Pope (1988) for the case of initiallysegregated scalarmixing in isotropic turbulence. Section 4 presents a summary and a brief discussion.

2 THE ENHANCED DIFFUSION MODEL

In this Section the enhanced diffusion model is developed for passive scalar mixing inisotropic turbulence.

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2.1 Theproblem

Consider the evolution of a scalar field t/> (y, t) in a constant-density, isotropic velocityfield u(y, t). The velocity field evolves according to the Navier-Stokes equation. Theinstantaneous evolution of the scalar field is governed by the equation

(1)

where D is the coefficient of Fickian diffusion. The scalar pdf evolves according to (Pope1985)

(2)

where 'I/; is the probability-space value of the scalar concentration and the conditionalscalar dissipation x( '1/;) is given by

x('I/;) = D( 8t/> 8t/> It/> = '1/;).By; Byj

(3)

The notation (blr) is used to denote the conditional expectation of b with respect tor. The conditional scalar dissipation is not closed in terms of the pdf and needs to bemodeled.

If the initial length scale of the scalar field is of the order of the velocity field lengthscales or larger, the effect of the turbulence is to cascade the scalar energy down to smallerand smaller scales until the molecular action dissipates the energy. This cascading effectrenders the modeling or direct calculation of the conditional scalar dissipation muchmore complicated than it would be in the absence of the turbulent velocity field. In theremainder of this Section, using a Lagrangian frame analysis and a simplification, theeffect of the velocity field is modeled as a time-dependent enhanced molecular diffusionoperating on the initial scalar field. .

2.2 Lagrangian frame analysis and development of the model

The effect of the velocity field is most easily analyzed in the Lagrangian frame of reference(x,t). Consider the evolution of the scalar field in a turbulent velocity field U(X,t), whereX represents a Cartesian Eulerian coordinate system. The X-coordinate of x evolvesaccording to

with the initial condition,

8X(x, t) _ U[X( ) 18t - x, t ,t ,

X(O) = x.

(4)

(5)

The velocity field U(X,t) evolves according to the Navier-Stokes equation. Now, let theLagrangian coordinate system x be Cartesian. In ·the coordinate system following fluidparticles, the scalar concentration evolves according to

8t/>(x,t) = Dfft/>(X,t),8t 8Xj8Xj

(6)

where D is scalar diffusivity. The molecular diffusion can be rewritten in terms of theLagrangian scalar derivatives:

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TURBULENT SCAlAR MIXING 87

(7)at/>(x, t) = D aXa aXb a2t/>(x, t) .

at aXi aXi aXaaXb

The evolution equation ofthe pdf, G(l/Jt, t), of the scalar concentration t/>(x, t) followingfluid particles in an isotropic field can be derived using standard methods (Pope 1985):

where

aG(..pI,t) = -D~[G(ol' t)(L L at/>(x,t)at/>(x,t)l-" = 01,)]at a..pla..pl '1'1, sa ib ax. aXb 'I' '1'1 ,

aXaL i n = - ·ex,

(8)

(9)

Due to the isotropy of the field, the Lagrangian scalar pdf G I (..pI, t) and the Eulerianpdf F (!/J, t) are identical.

What is L? Consider the evolution of material line elements in the turbulent velocityfield U(X,t). Let la(x,t) represent the line-element vector, at any time t ; of a materialline that is initially (i) located at x, (ii) of unit length, and (iii) aligned along the X a axis:i.e., la(x,O) = ea' The components of I, are l i«, 12a, and 13a. The evolution equation ofthe material line element is (Monin and Yaglom 1981)

ot., aUi(X(X, t), t)at Bx;

(10)

ou,= ax· l ja,

}

where X(x,t) is the location at time t of the fluid element initially located at x, Theevolution equation of X is given by equation (4). The initial condition on lia is

(11)

By differentiating equations (4) and (5) with respect to Xa , it can be easily seen thatthe evolution equation and the initial condition of Ii. and &X~(x,i) are identical. Hence

UX,

i.. = aXi(x, t).aXa

(12)

The material line element la is stretched (or shrunk), reoriented, and advected by thevelocity field. The line-element vector at any time t can be determined knowing thetensor B and the initial condition:

la(t) = B(t).ea . (13)

The tensor B contains all of the one-point information of the fluid element and evolvesaccording to (Monin and Yaglom 1981)

(14)

Clearly, Lai is the inverse matrix of lis- It can be easily shown that

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(15)L - Bx; Ca j

ai = 8X; = V'where Ca; is the i -th component of area of a surface element which is initially of unitarea and whose normal is initially coincident with the X a axis, and V is the volume (ordensity ratio) of the fluid element:

pet) f' 8U;Vet) = p(O) = exp[- 10 8X; (t)dt]. (16)

The scalar field is composed of isoscalar (concentration) surfaces. In the absence ofany velocity field, the scalar field diffuses due to the gradients in the direction normal tothe isoscalar surfaces. In general, the effect of the turbulent velocity field is to stretchthese isoscalar surfaces. As a consequence of incompressibility, this surface stretchingresults in isoscalar surfaces of different concentrations coming closer to one anotherresulting in a reduction of the scalar length scale. This transfer of the scalar energy fromlarger to smaller scales is the cascading effect of the velocity field. The net result of thevelocity field is, then, to steepen the gradients normal to the isoscalar surfaces which inturn results in faster diffusion of the scalar field. The greater the stretch in surface area,the faster the rate of diffusion and the quicker the scalar pdf evolution. This effect ofvelocity field on the scalar pdf evolution is reflected by the presence of the area-stretchterms La; and Lbi in equation (8).

The slmpllftcatlon. Consider the evolution of the scalar field from random initialconditions. Since the initial scalar field is randomly distributed, at early times (of theorder of a few Kolmogorov time scales), there will be little or no correlation betweenthe isoscalar surface areas (La;) and the scalar gradient normal to these surfaces (~.

At early times then, the correlation C between La; and B~:;t) is

(17)

With time, the isoscalar surfaces are strained and convected by the turbulent velocityfield. The evolution of the area of a surface element (followinga fluid particle) is dictatedby the small scales of the turbulent velocity field. Hence, the length scale of variationof Lai(x,t) at any time t can be expected to be of the order of the Kolmogorov lengthscale (1"/) of the turbulence. The length scale of B~:;I) at any time t in the Lagrangiancoordinates will be of the order of the characteristic small scale (1"/5) of the initial scalarspectrum. (Note that the scalar gradient in the Eulerian space gets steeper with time andthe length scale of 1J~~;t) would be of the order of the Kolmogorov scale.) The spatialcorrelation between these two quantities of length scales 1"/ and 1"/5 can be estimated tobe of the order of (Thnnekes and Lumley 1972)

8t/> r when ~<~,C(L,-)~

8X ~,/~ when ~,<~

!!.. when 1"/ « 1"/51"/5

(18)

(18)

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TURBULENT SCAlAR MIXING 89

7), h- w en 7) » 7),.7)

So it is clear that when 7) and 7), are not of the same order, the correlation between thearea of a surface element and the Lagrangian scalar gradient is poor.

Based upon the above arguments we hypothesize that to zero-th order

(ax. aXb aep(x, t) aep(x, t) lep = 'Pi) ::::: (ax. aXb )(aep(x, t) aep(x, t) lep = !/J,), (19)aXj ax; ax. aXb ax; aXj ax. aXb

where (£Jt~) is independent of the scalar field.Subject to the above assumption, equation (8) can be simplified as

Owing to the isotropy of the turbulent small scales and the initial scalar field, furthersimplification is possible:

(21)

where S, for reasons that will be apparent soon, is called the diffusion enhancementfactor. Then,

aG(!/JI,t) a2at ::::: -(S) a!/J,a!/J, [G(!/J"t)X' (!/JI)],

where XI (!/J,) is the Lagrangian conditional scalar dissipation defined as

2.3 The model

(22)

(23)

For the purpose of evaluating one-point statistics of a scalar field evolvingin a turbulentvelocity field, we suggest that turbulent scalar mixing equation (1) can be modeled bythe enhanced diffusion equation

aep = D(S(t»)~.at aXjaXj

(24)

In the above equation, (S(t») is the factor by which the Fickian diffusion coefficientis enhanced due to the presence of the turbulent velocity field and x is a cartesianLagrangian coordinate system. In modeling turbulent mixing as above, it is importantto note that the cascading effect of the velocity field is not neglected, but is merelydecorrelated from the diffusive effects of molecular action.

The instantaneous scalar dissipation implied by the model is

(25)

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The conditional scalar dissipation of the model is

(26)

The mean scalar dissipation is obtained by integrating equation (26):

(27)

where fl (t) is the mean scalar dissipation in an equivalent heat -conduction system:

8¢ 8¢fl(t)=D(-a s: ).

Xa Xa

The conditional scalar dissipation ratio, xlf" of the model is

x(1/J) xMd= (28)

and hence independent of the velocity field.From the form of the model, two important observations can be made. (i) As mentioned

earlier in this Section, the model can be expected to be reasonably accurate when theKolmogorov length scale of turbulence (7) and the characteristic small scale of the initialscalar spectrum (7)$) are of different orders of magnitude. It is surprising that neitherSchmidt number nor Reynolds number enter explicitly into consideration in this model.The objective of this paper is not to dwell on the Schmidt and Reynolds number effectsimplied by the model, but rather to test the proposed model against DNS results (ofEswaran and Pope 1988) and to demonstrate that the model does perform adequately.In fact, it will be seen that many of the features of turbulent scalar mixing that falloutside the scope of many of the present models (mapping closure model included)can be explained with the enhanced diffusion model. (ii) The diffusion enhancementfactor, like the Fickian diffusion coefficient, only alters the time scale of the scalar fieldevolution. Rescaling time as

T = l D(S(t»)dt, (29)

it is easily seen that the model pdf evolution in the time coordinate T is independentof the velocity field. This is in qualitative agreement with the DNS findings of Eswaranand Pope (1988).

The model is validated (quantitatively and qualitatively) against DNS data in the nextSection.

2.4 Evaluation of (S (t»)

The diffusion enhancement factor S (= ~~~) is a measure of the deformation of theLagrangian coordinate frame caused by the velocity field. It is given by (from equations21 and 15)

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TURBULENT SCALAR MIXING 91

where A~ = Lfa + qa + L~a' For an incompressible velocity field, the density ratio Vis always unity. The compressibility of the velocity field manifests itself on the scalarmixing through V. The implication is that the compressibility only affects the time scaleof scalar evolution without modifying the shape of the scalar pdf's. Indeed, Givi et al(1991) find that the scalar pdf shapes are not affected much by compressibility.

For an isotropic velocity field, the various area magnitudes (A I, A z, and A 3 are allstatistically equivalent. Hence, assuming isotropy (and incompressibility)

(S(t») = (Az(t»), (31)

where A (t) represents the area at any time t of any typical material-surface element (ofinitial unit area) associated with the fluid element.

A naive estimate of (S(t») that ignores the effects of vorticity and strainrate rotationcan be made (a La Monin and Yaglom 1981). Subject to the above conditions we canexpect

so that

A(t) ~ exp[(al + az)tj,

(S(t») ~ exp[2(al + az)tj,

(32)

(33)

where a1 and az are the means of the positive and intermediate eigenvalues of thevelocity gradient tensor. In real turbulence, however, the effects of vorticity and strain­rate rotation are important (Girimaji and Pope 1990).

Girimaji and Pope (1990) have performed extensive studies of the evolution of line,surface, and volume elements in statistically stationary, isotropic turbulence. One oftheir findings is that Ln (A) has a Gaussian distribution, which implies that A z has alog-normal distribution. Although no direct estimate of (S(t») was made, based on thefindings of that study, we can surmise some aspects of the behavior of (S(t»). Let usdefine

( )_ d In(S)

p t = dt . (34)

The exponential growth rate p(t) is a function only of the turbulence parameters and isvery likely to be positive at all times. In real turbulence, the temporal behavior of p(t) islikely to be very similar to the line and area logarithmic growth rates (Girimaji and Pope1990). At time t = 0, p(t) will be nearly zero, owing to the lack of alignment betweenthe material-element area ellipsoid and the velocity-gradient tensor. As the alignmentimproves with time, p(t) would increase, peaking at about tm = 5T~ (T~, Kolmogorovtime scale), when the alignment between the principal axes of the velocity-gradient tensorand those of the material-element ellipsoid is at the maximum. Finally, as vorticity andthe rotation of the strain-rate principal-axes reduce the alignment, p would asymptote toa lower value. Girimaji and Pope (1990) estimate the long-time growth rate of variousmoments of the surface areas from which p(t) can be estimated:

3 MODEL VS. DNS

p(t) = d In(A(t)Z) ~ 0.8.dt T~

(35)

In this Section, the model is computed against the DNS data of Eswaran and Pope (1988).First, quantitative comparison between the model and DNS is made for the evolution ofthe scalar pdf and the conditional scalar dissipation ratio. Then, it is demonstrated that

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the model explains the non-monotonic evolution of the mean scalar dissipation and thenear log-normality of the scalar dissipation pdf.

DNS data. In the direct numerical simulations, the scalar field evolves in a stationaryisotropic turbulent field from a segregated, symmetric initial pdf

F('ljJ.O) = ~[<5('ljJ) + <5('ljJ -1)J, (36)

where <5 represents the dirac delta function. The study was performed for a range ofTaylor-scale Reynolds numbers (RA = 30 - 50) and scalar field initial length scale(ls)'One of the main inferences from the DNS study is that at any given stage of pdf decaycharacterized by the variance (72, the shapes of the scalar pdf and the conditional scalardissipation are nearly independent of Is and RA.

Modelcalculations. Model predictions are calculated from numerical (spectral) solutionto the enhanced diffusion equation (24). Since, for the purpose of comparison here, thetime scale of evolution is irrelevant, the product D (S (t )) is taken to be unity for all times.The segregated initial scalar field is specified in a manner similar to that by Eswaran andPope (1988). The initial pdf is the same as that of DNS (equation 36). Since the scalarfield evolves according to the heat conduction equation, these simulations are called theheat conduction simulations (HCS). The HCS are performed for several values of Is.It is found that the pdf and the conditional scalar dissipation of the scalar at any givenvariance are nearly independent of Is (Girimaji 1992c).

3.1 Scalarpdf

We are interested only in the shapes taken by the pdf during its evolution and not thetime scale of the evolution. Hence, the comparison between the model and the data isperformed at certain given values of variance. At any given value of the variance, theHCS-pdf is independent of Isand the DNS-pdf is independent of I, and RA. Consequently,it is adequate to perform the comparison for anyone combination of Is and R A•

The results presented here are for initial scalar length scale /s = 21r18. In Figure 1,the scalar pdf's obtained from the HCS, the DNS of Eswaran and Pope (1988), and the,B-pdf model of Girimaji (1991a, 1991b) are compared at various stages of mixing. Whilecomparing the results, it is important to recognize that the initial spectral fields of theHCS and DNS are slightlydifferent. For the sake of better numerical resolution, the highwavenumber components of the DNS initial field are modified to yield somewhat smoothinitial fields. As a result, the scalar field is not completely segregated initially. The HCSinitial field is unsmoothed and the scalars are closer to being completely segregated. Thejaggedness in the early-time HCS pdf is due to the presence of the untruncated highwavenumber components of the scalar field. Given all these factors, the agreement isquite good. Both enhanced diffusion model and the ,B-pdf model capture the variousphases of the DNS evolution well. It has been established elsewhere (Girimaji 1992b)that for this initial condition the ,B-pdf model and the mapping closure model yield nearlyidentical results. Hence, it can be inferred that the enhanced diffusion model and themapping closure model are in reasonably good agreement.

3.2 Conditional scalardissipation ratio

The conditional dissipation ratio, x('ljJ)/f" plays an important role in the evolution ofthe shape of the scalar pdf, whereas the rate of the evolution is determined by f s. As

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TURBULENT SCAlAR MIXING 93

o

0.8 1.00.6

1.0

(72=0.1772.0

o

2.5

0.5

1.0

3.0

1.5

(72 = 0.073

(72 = 0.220

0.2 0.4 0.6 0.8 1.0

3.0

2.5

2.0Probability

density 1.5F(IjII

1.0

0.5

0

o

1.0

0.5

4.0

3.5

3.0

2.5

2.0

1.5

3.0 3.0

2.5 (72 = O.03G

2.0

1.5

1.0 1.0

0.5 0.5

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

FIGURE 1 Evolution of scalar pdf, F(I/J), for the case of I' = 0.5. Circles represent DNS data, dashedline represents HCS data, and solid line represents I'.pdf model calculations.

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94 S. S. GIRIMAJI

was seen in the previous Section, the conditional dissipation ratio of the present modelis independent of the velocity field. Hence, it is appropriate to compare the HCS datawith the DNS data. In Figure 2, the conditional dissipation ratios of HCS, DNS, and the,B-pdf conditional dissipation model of Girimaji (1992b) are plotted against normalizedmass fraction, (1/; - (</»)10-. The calculation of any high-order conditional statistic-likexlfs from numerical data is usually fraught with statistical errors. Taking this intoconsideration, the agreement between HCS and DNS is good for the early stage (0- =0.27) and adequate for the intermediate stage (0- = 0.1) of mixing. At the final stagesof mixing, the conditional dissipation ratio of the HCS data is nearly constant over therange of mass fraction considered. This is consistent with the fact that the HCS scalarpdf decays as a Gaussian at this range of variance. The DNS conditional dissipation ratiois minimum at 1/; = 0.5 and increases steeply for other values. This trend is inconsistentwith the observed Gaussianity of the scalar pdf and may be due to statistical error. TheHCS data comes from a 1283 simulation and, hence, can be expected to be statisticallymore accurate than the DNS data which comes from a 643 simulation. The ,B-pdf modelagrees well with both HCS and DNS data during early and intermediate stages of mixingand shows the same 'trend as the HCS data at the latter stages.

3.3 Mean scalar dissipation, f s.

From equations (27) and (34) the model mean scalar dissipation can be expressed as

(37)

In the DNS of Eswaran and Pope (1988) the mean scalar dissipation decays monotonicallyin time for small values of le]l«, where l« is the characteristic length scale of the energycontaining eddies. However, for larger Is values the mean scalar dissipation increases intime at the initial stages followed by a decay. Any quantitative comparison between theenhanced diffusion model and DNS is difficult, because the model for fs requires that(5 (t») be known. Can the non-monotonic behavior be explained, at least qualitatively,from within the framework of an enhanced diffusion model?

Equation (37) can be rewritten as

d Infs _ () d Infldt -pt+ dt' (38)

The Lagrangian dissipation fl (t) is a function only of the initial scalar field and decaysmonotonically in time: the rate of decay being larger for smaller Is and vice-versa. Thenon-monotonic behavior of fs in the DNS data must then be entirely due to the effect ofthe velocity field manifesting via the behavior of p( t). As seen in the previous Section,p(t) is non-negative in isotropic turbulence. Starting from zero at the early stages ofmixing, p(t) grows to a maximum value, peaking at tm(~ 5Tq ) , before settling down to asmaller asymptotic value (Girimaji and Pope 1990). If fs is to grow in time, we need itsgrowth due to p( t) to be greater than the decay due to lne.. This can happen for largeIs when the decay of fl is slow enough and when p(t) is near its maximum (t = tm).At later times, when p(t) settles down at its smaller asymptotic value, fs will decreasedue to the influence of fl. For the case of small L, the decay rate of Infl is always muchlarger in magnitude than p(t). Hence, for these values of Is> fs decreases monotonically.

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TURBULENT SCALAR MIXING 95

2.0

3

••

2

•

~.:g: -!'--.•A". -.

.... .... 0.... ........ 0 •

....,0,,....,

1o(0/ - ill/a

-1-20'----.......--""'-----"""-----'---.........-'----'·3

1.5 I::J.

I::J.

I::J.

I::J.

• I::J.1.0 . _._.-o-.Q

~ .......X(o/)/Es

.......0 ...

" •""""" •0.5 •

FIGURE 2 Conditional scalar dissipation ratio vs. normalized mass fraction for the case of IJ. = 0.5.Comparison of DNS data, HCS data, and !J-pdf model (Girimaji 1992). (0' = 0.27: 0 DNS,. HCS,-- model. 0' = 0.10: 0 DNS, mICS, - - - model 0' = 0.02: 6 DNS, • HCS, -- . -- .-- model.)

3.4 Pdf of scalar dissipation

Eswaran and Pope (1988) find that at long times, the pdf of scalar dissipation is log­normal in the presence of isotropic turbulence. We now investigate if our present analysisof the role of the velocity field can predict log-normality of the scalar dissipation.

Let 11(= it it) be the instantaneous value of the scalar dissipation. As before, wecan write

(39)

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%

where

s. S. GIRIMAJI

(/ -!!.t {},p nd T _ Cia Cibab - Bx; {}Xb a ab - V V·

The matrix T is symmetric and positive definite. At long times, it would not beunreasonable to expect the relative orientation of the principal axes of T and the tensorO~b to reach a stationary state. Then, at long times,

OCt) ~ IT(t)IIO' (t)l,

where IT I and 10'1 are the norms. This can be further simplified as

(40)

(41)

where A(t) again represents the area of a typical material surface associated with thefluid element. The physical meaning of this approximation is that at long times, thesmallest length scale associated with a material element is the volume (of the fluid­element) divided by the area of a typical surface element. The scalar dissipation, beingthe square of the scalar gradient, should be inversely proportional to the square of thesmallest length scale. So, at long times,

In(IOI) ~ In(lOI) + 2In(A) - 2In(V). (42)

In incompressible turbulence, V (t) = 1 and, hence, In(V) =O. The only fluctuations in 0are then due to fluctuations in 0 and A. As mentioned before, in isotropic turbulence, thearea of a surface element is log-normally distributed. At long times the fluctuations in 0,are likely to be small, for in a pure molecular diffusion case the scalar gradient decreasesmonotonically in time. Hence, the pdf of scalar dissipation at long times, implied by themodel, should be close to lognormal, modified slightly due to the fluctuations in 0,.

A somewhat similar explanation for the lognormality of scalar dissipation is providedby Hill and Bowhill (1978) who neglect the effect of molecular diffusivity. The presentanalysis is more complete in that (i) the role of molecular diffusivity is better identified,and (ii) many of the other DNS observations, which are beyond the scope of the previouswork, are explained.

3.5 Extreme valuesof the scalar

For the turbulent mixing problem on hand, it has been shown in Girimaji (1992c) thatthe extreme values that the scalar takes with non-zero probability change with time,converging to the mean value at long times. The mapping closure model predicts thatthe extreme values remain fixed at the initial extreme values. This leads to a formalfailure of the mapping during the latter stages of mixing (Girimaji 1992a). However, inthe heat conduction calculations the extreme values do change with time, asyrnptotingto the mean value. Hence, at least qualitatively, during the latter stages of mixing, theenhanced diffusion model is more consistent with the physics of turbulent scalar mixingthan is the mapping closure model.

4 SUMMARY AND CONCLUSIONS

An enhanced diffusion model to simulate the evolution of scalar pdf in isotropic turbulenceis developed. Many of the aspects of scalar pdf and conditional dissipation observed in

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TURBULENT SCALAR MIXING 97

DNS of Eswaran and Pope (1988) are captured quite well by the model. Most of thepresent turbulent mixing models (e.g., mapping closure model, t3-pdf model) shed nolight on the role of velocity field in scalar mixing. Unlike these models, the present modelquantifies the effect of the velocity field on mixing in terms of the material deformationcharacteristics of turbulence. Quantitative validation of these model predictions havenot been performed in this paper due to the lack of relevant DNS Lagrangian data.However, the qualitative trends implied by the model regarding the role of the velocityfield are in good agreement with pNS observations. In this sense, the enhanced diffusionmodel is more complete than the mapping closure model.

An examination of the model's validity range (in terms of the Schmidt and Reynoldsnumbers) falls outside the purview of the present effort. It is clear from the goodagreement between the present model and DNS that there does exist an important rangeof these parameters where the model is valid. This model will be valid in the range ofparameters where the mapping closure model and the t3-pdf model are applicable, forthose models are also not sensitive to Schmidt and Reynolds numbers.

The HCS methodology developed in this paper has already been used to study severalaspects of turbulent scalar mixing (Girimaji 1992c). The HCS have been particularlyuseful in exposing some of the shortcomings of the mapping closure model during thefinal stages of mixing and ultimately identifying the exact reasons for these failures(Girimaji 1992a, 1992b). The enhanced diffusion model, further, does not suffer fromthese shortcomings during the final stages of mixing. The HCS can also be used toaddress several issues regarding chemical reactions in turbulent flows. Since chemicalreaction is a process that is local in space, several of the processes that need to bemodeled in turbulent combustion are also local in space and, hence, can be investigatedin the absence of the velocity field. A study of a random scalar field of several speciesundergoing molecular diffusion and chemical reaction willbe useful in assessing the effectof reaction on concentration fluctuations, the importance of temperature-concentrationcorrelations, etc.

ACKNOWLEDGEMENTS

1 would like thank Dr. J. P. Drummond for his encouragement and support. This work was supported bythe Theoretical Flow Physics Branch, Fluid Mechanics Division. NASA Langley Research Center, Hampton.VA 23681, under contract No. NASI-18599.

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[4] Gao, F. (1991). An analytical solution for the scalar probability density function in homogeneous turbulence.Phys. Fluids A, 3 (4), 511-513.

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