Engineering Turbulence Modelling and Experimcnts - 5

59W. Rodi and N . Fueyo (Editors )© 2002 Elsevier Science Ltd . All rights reserved .

TURBULENT MIXING AN D

COMBUSTION MODELLIN G

Jesús Martín 1 and César Dopazo 2

'Fluid Mechanics Group. University of Zaragoza . Zaragoza 50015 . Spain .

2Center for Energy and Environmental Research (CIEMAT) . Ministry of Science andTechnology . Madrid 28040 . Spain .

1 Introduction .

Modelling the turbulent mixing of reacting scalars remains a challenge . The difficulties in de-scribing mathematically the turbulence, the main driving mechanism for scalar field large-scal econvection and small scale strain and rotation, are compounded with the molecular transpor tand non linear chemistry interactions . In turn, the turbulent vélocity may be significantl yaffected by the thermal patterns generated by the chemical reaction or combustion proces sof gaseous fuels . Liquid and solid fuel turbulent combustion is even harder to tackle, sinc eprocesses such as atomization, evaporation, devolatilization and heterogeneous must be als o

incorporated into the models .This paper will be restricted to review ideas on the turbulent mixing of a single react-

ing scalar in a statistically homogeneous constant density flow . Some of the basic problemsin modelling turbulent combustion can be illustrated with this simple approximation . Thehandling of complex chemical kinetics or reduced chemistry models for multi-scalar fields, a swell as the thermal effects upon turbulence due to exothermal reactions have been examine delsewhere[11,[2]

Very few conclusive facts are known about turbulence [3] , not to speak on the inclusion o f

reacting scalar fields . This, combined with the peed to provide urgent answers, such as thos erelated to the improvement of combustion efficiency, stability and noise/vibration generation a swell as reduction of pollutant formation and emissions, has caused the proliferation of aprioristi cphenomenological models that, while providing practical guidelines, help very little in gainingsome basic knowledge of the overall processes .

6 0

Emphasis in this paper is placed on the rigorous use of the scalar conservation equation . InSection 2 the main transport equations are presented and the velocity field small scale topologie sare described. In this frame, correlations between the scalar and velocity fields, resulting fro mnumerical experiments, are discussed . Section 3 introduces some notions of dynamical systemsin the scalar mixing problem. The mixing of a reacting scalar is considered in section 4 .

2 Effect of the small-scale flow topologies on scalar mix -ing .

The statistics of a passive scalar in turbulence has been widely investigated in recent years usin gdirect numerical simulation (DNS) of the Navier-Stokes equations and the transport equationof an inert scalar (See, for example, Kerr í *, Ashurst et a1 . [5] ; Ruetsch and Maxey [6] or Pumir [7] ,among others) .

Scalar mixing in a turbulent flow is characterized by multiple length and time scales ove rwhich scalar fields significantly fluctuate. Along this process a variety of physical mechanism stake place (see Ottino [8 and Dopazo [9] ) ; apart from mean and large scale convection, th escalar field evolves under the random straining and rotation created by the turbulence ; scalarheterogeneities are smeared out by molecular diffusion enhanced by stretching and folding o fisoscalar surfaces . Turbulent mixing with reduction of scalar fluctuations has its counterpartin the turbulent kinetic energy decay process, the latter being also influenced by the pressur eisotropizing action .

The rational formulation of stochastic molecular mixing models crucially depends on th eability to parameterize the previous mechanisms in terms of the scalar fluctuations, the scala rgradient vector, and knowable information pertaining to both the scalar and the turbulenc efields . It is, thus, logical to search for some correlation patterns between molecular transpor tterms in the conservation equations of the scalar related magnitudes and different propertie scharacterizing the velocity field .

Chong et al . [10] introduced a new methodology to characterize the small scale topologiesof turbulence motions in tercos of the three invariants of the velocity gradient tensor . Theseinvariants completely determine the nature of the dynamical system associated to a local origi nwhich, in a Langragian frame, is a critical point and describes the small scale fluid motions a teach point of the domain . Analogous invariants can be defined for the strain rate tensor an dthe rate of rotation tensor, the latter related to the vorticity . Soria et al . [" investigated thenature of plane turbulent mixing layers with different initial conditions using DNS and analyzedthe velocity gradient tensor and strain rate topologies, finding highly localized regions for thesamples in the resulting scatter-plots of the invariants .

2 .1 The scalar field . Main variables and definitions .

Let C - C(x, t) be a chemically inert, dynamically passive and statistically homogeneous scala rfield. The mean value < C > is a constant . The scalar fluctuation c, defined as c = C— < C > ,obeys the equation

at + u . áx = DV2 c .

(1 )

D is the Fickian diffusion coefficient of c in the mixture and u is a zero-mean randomsolenoidal, statistically homogeneous and isotropic velocity field, governed by the Navier-Stoke sequations .

6 1

The dynamics of the scalar gradient vector, c, i , is essential to the scalar field evolution . Thisvector is perpendicular to the local isoscalar surface, at each point in the flow ; the geometr yand characteristics of these isosurfaces are determined by the spatial distributions of the scala rgradient [8I . The transport equation for c, i i s

ac,i

ac,i

2

at +ujax = —c,j uj,i+ DV c,i .

(2 )

The first term on the right hand side of this equation accounts for the local scalar gradien tstraining and rotation by the action of the velocity gradient tensor uj,i . The second term is thescalar gradient diffusion .

The time rate of change for the scalar variance, < c2 >, can be easily derived from Eq . (1) ,resulting in

d<c 2 > =— 2 <E,> .

(3 )d t

where E, = Dc, i c,i is the local and instantaneous scalar fluctuation dissipation rate . Thismagnitude, essential in the scalar mixing process, plays a role for the scalar heterogeneitie sanalogous to that of the kinetic energy dissipation rate, e, for the turbulent kinetic energy. Thetransport equation for e, is readily derived from Eq . (2), obtaining

at+ ui

a*c

-2Dui,jc,i

+D \7 2 E, — 2D2c,ij c,ij .

(4 )7

The first term on the right hand side of Eq . (4) is the production of E c by straining o fscalar gradients . Notice that rotation is absent from this term, since u i,j may be replaced by it ssymmetric part, the strain rate tensor, Sij ; it is analogous to the production term Sijw i wj in thesquared-vorticity (w 2) equation . The second term in (4) is the molecular diffusive transport ,and the third, is the dissipation of e* .

The E,-production term can be written as -2Du i,jc, i c, . = - 2Q Vc Ec , where the magnitud eQv c is defined as

Si,jc,ic , j

o-ve

( 5 )C,k C, k

Qvc is the scalar gradient stretching rate, and it can be described as the part of the strainaligned with the scalar gradient vector, analogously to the vorticity stretching rate introduce dby Jiménez et a¿[121 in the study on turbulent vorticity dynamics .

2 .2 Velocity gradient invariants . Small scale topologies .

A useful tool to describe and classify the small scale patterns of motions in a turbulent flo wis the Topological Methodology introduced by Chong et al. [10] . In this approach, for a constantdensity flow, the local velocity field dynamics at each point is characterized by two quantities :R and Q, the second and third invariants of the velocity gradient tensor = au i /axj . Theirexpressions are given by

Q = 1_ 2Ai3Aji (6)

1R = -3Ai - AjkAki (7)

62

Four different non-degenerate topologies exist in an incompressible flow : Stable focus withstretching, (SF/S), unstable focus with contraction, (UF/C), unstable node-saddle-saddle ,(UN/S/S) and stable node-saddle-saddle, (SN/S/S) . This classification is represented in Fig-ure 1 . The null discriminant curve, also displayed, separates focal, vorticity dominated, motion s(aboye) from dissipative, strain dominated, motions (below) . Focal motions are characterizedby positive values of the second invariant, Q, while dissipative motions are associated wit hlarge negative values of Q. Positive values of the third invariant, R, indicate two-dimensionalextension with contraction in the third dimension, while R negative implies one-dimensionalstretching with contraction in a plane .

Q

o

Figure 1 : Topological classification of turbulent small scale motions in terms of the velocit ygradient tensor invariants R and Q. The null discriminant fine is also shown .

Analogous invariants can be defined for the symmetric and skew-symmetric parts of th evelocity gradient tensor, the strain rate tensor Sij = (Aij + Aji)/2 and the rate-of-rotationtensor Wij = (Aij — Aji)/2, respectively. The non-zero second and third invariants of thes etensors are Qs, Qw and Rs, defined as

2WZ'W'i

( 9 )

1RS = - 3 Sij sjkski

(10)

Qs, always negative, is proportional to the kinetic energy dissipation rate E, namely E =

-4vQs . Qw, always positive, is proportional to the squared-vorticity, often called enstrophy ,

Qs =

( 8 )

Qw =

6 3

by Qw = w2/4. The invariant Rs characterizes the local geometry of the strain : a positive Rsvalue implies two-dimensional extension with contraction in the third dimension, and negative

Rs indicates the opposite .The invariants R and Q can be expressed in terms of Qs, Qw, Rs and the enstrophy

production term, Stiiw iw;, as

Q = Qs+Qw ,

R = Rs — 4S,,wzw3 .

These equations show that Q is the resulting balance between the local strain (Qs) and th e

local vorticity (Qw) magnitude. The value of R is determined by both the strain geometry an d

the enstrophy production term .The evolution of velocity gradient invariants in isotropic turbulence, using DNS, has been

investigated recently by Martín et al . [131 and Ooi et a1. [141 . Figure 2 show the joint statistic sand dynamics resulting from DNS in the planes R — Q and Qw — (—Qs) . Points in the phaseplane R — Q move spirally in a clockwise rotation around the origin, and approximating it in

a sequence SN/S/S — > SF/S — > UF/C — > UN/S/S . Points in the plane Qw — ( — Qs )move upwards on the left reaching at the beginning relatively larger values of strain ; then theenstrophy increases at a higher rate, until it dominates over the strain ; finally the magnitud eof both invariants are reduced, moving the points towards the origin along paths close to th eQw-axis, with enstrophy dominating over the strain .

(11 )

(12)

Qw

oN0

Figure 2 : Joint pdf and dynamics in the planes R — Q and Qw — ( — Qs)

2 .3 Numerical experiments and results .

The results presented here have been obtained from the numerical simulation of an inert scala r

in forced isotropic turbulence . The mesh consists of 128 3 points, with a Reynolds number

Rea 47. The Reynolds number has been deliberately kept at a low value to ensure the high

numerical resolution necessary to calculate high order scalar spatial derivatives . The scalar i s

bounded in the interval [0, 1] . The initial scalar distribution used is a double Dirac delta wit h

the peaks at the extreme values, 0 and 1 . Initially, the region in the cube center presents a

value C = 1 and the rest C = O . The scalar mean is 0 .5 . The Schmidt number is Sc = 1 . 0

since the kinematic viscosity v and the scalar diffusivity D have the same value 0 .025 . Results

6 4

presented in this section correspond to a time when the velocity field has reached a statisticallystationary state and the scalar probability density function (pdf) is almost identical everywher ein physical space .

2.4 The scalar field .

In order to obtain a spatial picture of the resulting scalar and velocity fields, graphics showin gisocontours of some variables in a plane of the simulation cube have been depicted . Figure3 shows isocontours of the squared scalar fluctuations and the scalar dissipation rate . Eachmagnitude has been made dimensionless with its average over the whole domain . The scalarfield appears as an spatial distribution of large 'blobs' with no particular structure . Organizedstructures of intense scalar dissipation appear as ribbons surrounding the borders of the scala r'blobs' in thin, strained regions . This indicates that strong scalar dissipation structures arespatially organized as border sheets separating the 'blobs ' . Figure 4 shows, at the same plane ,isocontours of the invariants R and Q, which have been made dimensionless with adequatepowers of the averaged enstrophy < Qw > . The velocity field presents a higher complexity .Intense vorticity structures (large positive values of Q) appear isolated and surrounded in it svicinity by strained structures (negative values of Q) which show a similar level of intermittency .R invariant generally shows opposite sign values to those of Q in those regions of the plan eassociated to organized structures, as expected from previous results in the literature describin gthe joint statistics of both invariants .

squared sca la r fluctuation

scalar dissipation rate

Figure 3 : Isocontours of the squared scalar fluctuations c 2 and the scalar dissipation rate e, ina plane of the simulation cube . Both magnitudes are nondimensionalized by their averages .

The pdf of the scalar dissipation rate E, is illustrated in Figure 5 . It is apparent an apprecia-ble probability of very high values of E, up to twenty times its average, 0 .07 . This intermittentbehavior has been widely described in the literature on the subject . The growth of the scalardissipation has its origin in the amplification of scalar gradient vectors due to large velocitygradients occurring in the small scales of the velocity field . This effect can be analyzed throug hthe scalar dissipation production terco, PES = -2Dui j c, i c,i , of equation 4 . The pdf of thi squantity is also displayed in Fig. 5. The result shows a strong intermittent distribution, a sexpected for a triple product of derivatives, with the part corresponding to positive values cov-ering a significantly larger range than that for negative ones . PE , presents a positive average o f0.5 with a standard deviation of 1 .57 and a large positive skewness factor of 6 .7 . The enstrophy

65

R

Q

Figure 4: Isocontours of the invariants R and Q in a plane of the simulation cube . The invariantshave been nondimensionalized with the mean enstrophy, < Qw > .

production term, Sijwiwj , is, in the average, positive. This well known property is explained bythe alignment between the vorticity and the principal axis associated to the (mostly positive)intermediate strain eigenvalue /3[5] . A similar argument is used to justify the negative sig ngenerally encountered for Sij c, i c, j , since the scalar gradient vector shows preferred alignmen twith the axis corresponding to the negative strain eigenvalue ry . This fact explains the positiv eaverage found for the dissipation production term, -2DSijc, i c, j .

á 0Q.

0 .2 0.4 0 .6 0.8 1 1 .2 1 .4 1 .6 1 . 8

100

1 0

1

0 . 1

0.0 1

0 .00 1

0 .0001 :

le-050

10

1

0. 1

0 .0 1

0.00 1

0 .0001 :

Tle-05

le-06

-10

-5

0

5

10

15

2 0

scalar dissipation

scalar dissipation production term

Figure 5: pdfs of the scalar dissipation cc and the scalar dissipation production ter m—2DSi,.7c,i c,j .

2.5 Correlations with the velocity gradient invariants .

Correlations between scalar mixing and the turbulent velocity field local properties from atopological perspective are studied through the averages of the scalar field main variables con-ditional upon the velocity gradient invariants, which are significant quantities describing th evelocity field characteristics . Invariants have been nondimensionalized with the mean enstroph y< Qw >, while scalar related magnitudes are nondimensionalized by the pertinent averages .Figure 6 shows the curves of the square scalar fluctuation c 2 conditional upon the invariants Rand Q and then upon the invariants —Qs and Qw. Conditional averages of the scalar dissipa-tion rate cc upon the four invariants are displayed in Figure 7 . c2 presents the most significant

66

correlation with the invariant Q, being the variance decreased until 2/3 of the overall valu ewhen larger negative values of Q are involved . The dependence of the variance over the restof invariants is weaker . On the contrary, the scalar dissipation E, presents strong correlationswith R, Q and Qs . E, remains approximately constant in the range of negative values of R,and increases as large positive values of R are reached . The dependence oil Q is even stronger ;the scalar dissipation reaches values of more than three times its average for extreme negativ evalues of Q . The behavior of E, in regard to the strain is clearly different from the behavio rrespect to the vorticity, since E, takes larger values when —Qs increases, while showing ver ylow dependence on Qw .

upon Rupon Q ----

8

o-4

2

1 .5

0 .5

2

1 . 5

oJ-oo1 0

o0

8 0 .5

o3-3 -1-2 o 22 o4 3 54 6 7

Figure 6: Square scalar fluctuations, normalized by the scalar variance, conditional upon th einvariants R, Q and also upon —Qs, Qw . The invariants have been nondimensionalized wit hthe mean enstrophy, < Qw > .

Figure 7 : The scalar dissipation rate, normalized by its average, conditional upon the invariantsR, Q, —Qs and Qw .

Figure 8 shows conditional averages of the scalar dissipation production term in Eq . 4 asa function of the invariants . Here, the vertical axis has been nondimensionalized with the roo tmean square of the scalar dissipation time derivative over the total flow, as the dissipationproduction is one of the contributions to that time derivative . Looking at the graphics, thestrongly intermittent dissipation production term results in averages up to six times the overal laverage for extreme positive values of R and also for extreme negative values of Q . On the otherhand, the dissipation production strongly increases with increasing —Qs values, while oppositebehavior is found respect to the enstrophy values, since only a slight steady decreasing of th eproduction as Qw increases is observed in the corresponding graphic .

67

upon Rupon a

-3

-2

-1

0

1

2

3

4

Figure 8 : The scalar dissipation production term, normalized with the rms of dec/dt, condi-tional upon the invariants (a) R, Q, —Qs and Qw .

These last results indicate that scalar variance has slight direct dependence on the smal lscale structures. On the contrary, scalar dissipation presents correlations with R and Q. Thecorrelation with Q comes from the strain part, —Qs, with very weak dependence on the vorticity ,Qw . The dissipation production term presents analogous -though stronger- trends respect t oR, Q and Qs, and very weak dependence on vorticity.

The scalar variance presents weak correlations with the small scale topologies of the tur-bulent velocity field, while both the scalar dissipation and its production term are dependenton these structures . This fact can be explained in terms of appropriate characteristic times . Acharacteristic variation time for a variable X can be estimated a s

te(X) = Xrlma

(13 )( Dt J rms

where the symbol rms stands for the root mean squared of the quantity in brackets . Char-acteristic variation times have been obtained for the scalar fluctuations : t c (c) = 0 .60, thescalar dissipation : tc (€c ) = 0 .14, and also for the four invariants : t c(R) = 0.09, tc (Q) = 0 .13 ,t c (Qs) = 0 .11 and tc(Qw) = 0.21 . Characteristic lifespan times of topologies are defined byt c (R) and t c (Q), which are of the same order than t,(e,), but three times smaller than th escalar fluctuations time t c (c) . Correlations between scalar dissipation and invariants are thenjustified . In contrast, the scalar variance decay occurs in characteristic times three times large rthan the typical times of variation of R, Q and Qs invariants, so the kind of small scale motionpattern -topology- associated to a fluid particle changes faster than its scalar value, and so th elocal scalar variance results weakly correlated with the local topology .

2 .6

Scalar statistics in the four topologies.

Once the correlations of the main variables pertaining to the scalar field with the velocit ygradient invariants R and Q have been established, it seems logical to investigate the scalarfield behavior in the four topologies defined by both invariants. The statistical calculationswere performed over a total number of samples N = 2, 097,152 (= 128 3 mesh points in th esimulation cube) . The percentages of samples corresponding to each topology are 39 .8% forSF/S, 27.6% for UF/C, 25.5% for UN/S/S and 7.1% for SN/S/S. Table 1 shows the average sof the main variables in this study . Apart from magnitudes already introduced, the three strain-rate eigenvalues (a, Q and y) and the cosines of the angles between the scalar gradient vecto rand the strain-rate principal axes (i a , i fi , i,7 ) are also included . Averages of all magnitudes have

68

been calculated in the four topologies and finally in the whole flow for comparison . The variance< c2 > is roughly the same in the four topologies, but the averaged scalar dissipation obtaine dfor the UN/S/S topology, 0 .10, results about a 50% higher than the total average, 0 .07. Theproduction of dissipation and the scalar gradient stretching appear also strongly enhanced i nUN/S/S structures, being the average values of both quantities in this topology about twic ethe total averages . The scalar gradient presents strong alignment with the principal axis o fthe strain i7 associated to the negative eigenvalue ry . This effect is even higher in the UN/S/ Stopology, where the averaged cosines of the angle between i7 and Vc reaches a value of 0 .76 .

SF/SD>O,R<0

UF/CD>O,R>0

UN/S/SD<O,R>0

SN/S/SD<O,R<0

ALLSAMPLES

-Qs 26.2 24.6 46 .0 29 .1 31 . 0Qw 46.5 28.3 14 .5 13 .4 31 . 0

Siiw iw* 395 .6 -34.8 178.1 131.6 202 .5a 4.2 3 .9 5 .1 4 .8 4 . 4

0 .7 0 .9 1 .9 0 .3 1 . 1ry -4.9 -4.8 -7.0 -5.1 -5 .5c2 0.11 0.11 0 .10 0 .11 0.11E* 0.06 0.05 0 .10 0 .07 0.07

-2DSi;c, i c,* 0 .27 0.26 1 .13 0 .45 0.50Qvc -1 .31 -1 .05 -3.41 -1 .82 -1 .81

cos0a 0 .43 0 .48 0.36 0 .39 0 .42cosO fi 0 .40 0 .40 0.31 0 .39 0.38cos87 0 .65 0 .62 0.76 0 .69 0.67

Table 1. Averages of the main variables calculated for each topology, and also for the whol eflow .

The contribution, in percentage, coming from each topology to the integrated variable i nthe whole flow has been obtained for the strain, vorticity, scalar variance, scalar dissipationand dissipation production . The result in shown in Table 2 . The largest fraction (37 .8%) of th etotal strain -Qs -that is the total kinetic energy dissipation- comes from UN/S/S topologies ,followed by the SF/S topology contribution (33 .6%) . On the other hand, most of the enstroph y(Qw) in the flow occurs in SF/S structures (59 .7% of the total) . The scalar variance fraction sare approximately equal to the respective fractions of samples in the four topologies as expecte d(similar scalar variances resulted for all the topologies) . The scalar dissipation contribution sare very close to the ones obtained for the invariant -Qs, which indicates that intense scala rmixing is correlated with kinetic energy dissipation process . This is clear, looking at the resul tfor the production term of c c : More than half of the total scalar dissipation production, 57.8% ,has its origin in UN/S/S structures, while only a fraction of 21 .2% occurs in SF/S structures .

The probability density functions (pdf) of the scalar fluctuation c, the dissipation E, andits production term -2DSi;c, i c, i have been calculated in the four topologies, and the resul tis depicted in Figure 9 . Scalar dissipation and its production term have been normalized bytheir rms. While the scalar fluctuation distributions present similar curves, the result for E *

is quite different, showing that the probability of the highest values of the scalar dissipation i sabout ten times larger in UN/S/S topology than in the others . The distributions correspondingto focal, vorticity dominated topologies (SF/S and UF/C) result with smaller probability o flarge dissipation values . Similar trend, but stronger, is found for the dissipation productio nterm. The four curves on the right semiplane are clearly separated, having the one associate d

6 9

SF/SD>O,R<0

UF/CD>O,R>0

UN/S/SD<0,R>0

SN/S/SD<0,R< 0

% samples 39 .8 27.6 25 .4 7 . 1

—Qs 33 .6 21 .9 37 .8 6 . 7Qw 59.7 25.3 11 .9 3 . 1c2 40.3 28.7 23 .9 7 . 0E c 33.8 21 .5 37.3 7 . 3

—2DSi1 c, i c, i 21 .2 14.5 57.8 6 .5

Table 2 . Contribution (in percentage) of each topology to the integrated magnitude in the flow .

to planar strained topologies, UN/S/S, the largest probability of large positive production ,followed by SN/S/S, UF/C and SF/S topologies, in that order .

SF/SUF/C —-

UN/SIS ..SN/S/S

1

-0.4

-0 .2

0

0 .2

0. 4

scalar fluctuation

1 0

0. 1

0 .01

2

4

6

8

10

1 2

scalar dissipation

1 0

a-

-4

-2

0

2

4

6

8

1 0

scalar dissipation productio n

Figure 9 : pdfs of the scalar fluctuation c, the scalar dissipation rate E c and the scalar dissipationproduction terco -2DSt,ic, ti c, i calculated in the four topologies .

Conditional scalar diffusion < DO 2cic > and conditional scalar dissipation < Ec Ic > func-tions have been widely investigated in scalar mixing modeling . In the present work, thes efunctional dependences have been calculated considering the different small scale topologies .The results are displayed in Fig . 10. The curves corresponding to strained topologies presen tlarger slopes and reach higher maximum values than the other ones . Both conditional diffu-sion and conditional dissipation show the largest amplitudes along the whole scalar fluctuatio ndomain in UN/S/S structures .

70

SF/SUF/C —-

UN/S/S

SN/S/S

0 .6

0. 4

0 .2 5

0 . 2

0 .05

0-0.4 0 .4 -0 . 4-0.2

0

0 . 2

scalar fluctuation

-0 .2

0

0 . 2

scalar fluctuation

0 . 4

Figure 10 : Conditional scalar diffusion and conditional scalar dissipation calculated in the fou rtopologies .

2 .7 Joint conditional averages on the invariants R and Q .

It has been shown throughout the previous results that scalar mixing intensity, in particularscalar diffusion and scalar dissipation processes, are correlated with the topological structure sof the turbulent velocity field . In order to have a more precise picture of these dependences ,joint conditional averages of scalar related magnitudes upon R and Q, the -nondimensional-invariants, have been calculated from our DNS data . The R — Q plane has been divided i nrectangular cells, and the domain shown in the graphics is restricted to cells containing asignificant number of samples to achieve stable statistics .

Averages for square scalar fluctuations and scalar dissipation, normalized by their respectiv etotal averages, conditional upon R and Q are shown in Figure 11 . c 2 presents little variationin the domain, reaching the highest values for large positive Q, and smaller ones at the regio nbelow the D = 0 curve, corresponding to strained topologies . The averaged scalar dissipatio nshows instead a wider range of variation . Starting in the first quadrant, and following a nanti-clockwise rotation, e, is continuously increased, with the largest values occurring far fint othe fourth quadrant . The larger the distance to the origin, along the right branch of the nul ldiscriminant curve, the larger the values of the scalar dissipation rate .

Scalar mixing intensity is enhanced in high strain regions of turbulence, being this effec tstronger in UN/S/S small scale topologies, associated to planar strain . The averaged scalardissipation rate E, reaches its maximum values in this type of topological structures . Theterm responsible for the growth of scalar dissipation in equation 4 is -2DSi,i c, i c, i . Using thedefinition of the scalar gradient stretching rate given in Eq . 5, this production term can b ewritten as -2DSi,jc, i c, i = -2av 5 E c .

Conditional averages upon R and Q of these quantities concerning the production of scalardissipation, have been calculated . Fig . 12 shows, on the left, the result for the scalar dissipatio nproduction term, normalized with (d€ c/dt)rms , the rms of the dissipation total time derivative .Negative mean production is found solely in cells of UF/C topologies, while positive productio noccurs in the rest of the domain with large values in strain dominated topologies, being thes evalues increased for SN/S/S topologies far from the origin, along the region below the nul ldiscriminant curve . The conditional scalar gradient stretching av c , normalized with the strainmagnitude < Q,s > 1/2 , is depicted on the right of the same figure . It behaves similarly to theproduction, but has a less steeper increasing along the part of the domain surrounding thenull discriminant line. The averaged production value is about six times higher at the extrem eof the SN/S/S region than its value at the origin, while the scalar gradient stretching valu eresults only augmented three times between these points . The scalar dissipation resulted, for

7 1

-1

-1

b

i

á

-i

ó

1

áR

R

Figure 11 : The squared scalar fluctuation and the scalar dissipation, nondimensionalize dwith their averages, conditional upon R and Q . Both invariants have been nondimensionalizedwith the mean enstrophy, < Qw > .

extreme planar topologies, with a value two and a haif times the value at the origin . Thedissipation production term is proportional to the product of the scalar gradient stretching an dthe dissipation, as seen before . These results indicate that dissipation production comes fromlarge amplitudes of both cc and o-v, simultaneously, in a nonlinear way .

1 productlo nof dIssipatio n

term

sca lar gradien tstretchlnq rate

Q

ó

i

2

á

2

i

ó'

' 1' óR

R

Figure 12: The scalar dissipation production term and the scalar gradient stretching rate uv cconditional upon R and Q

In order to obtain a more detailed picture of the scalar dissipation dynamics, we hav ealso calculated, in addition to the scalar dissipation production, conditional statistics of th ediffusion and dissipation terms in the right hand side of Eq . (4) . Figure 13 shows averages ofthe three terms conditional on Q, and also their sum : the total time derivative of E c . The resul tis nondimensionalized with (d€c/dt) rms , and the invariant Q with < Qw > . The total timederivative of the scalar dissipation, averaged over the total simulation domain, results negativ e(-0.077) from our data . Looking at the figure, negative values of dEjdt are found for positiveQ and also for small negative values of this invariant . However, positive averages of the time

72

derivative are obtained in the range of moderate and large negative values of Q . Looking at th ebehavior of the different terms, the diffusion term is relatively small in the whole range of Qvalues and present opposite sign to the total time derivative . On the other hand the production -always with positive average- and the dissipation -negative definite- result strongly enhanced fo rlarge negative values of Q, being both of them drastically reduced when Q is positive . So scalardissipation growth results from a balance between large (positive) production and a slightl ysmaller (negative) dissipation in regions of the flow with large strain . When vorticity dominatesover the strain, the dissipation dominates over the production, resulting in a reduction of th escalar dissipation .

o

Figure 13: Averages of the three terms in the scalar dissipation equation (Eq. 4) conditionalon the invariant Q . The averaged total time derivative dec/dt conditional on Q is also depicted .The three terms and the time derivative have been nondimensionalized with (d€c/dt)r,,Ls, andthe invariant Q with the mean enstrophy, < Qw > .

3 Scalar and velocity field evolution . Averaged dynam -ical systems .

A further step in the study of a scalar in a turbulence field from a dynamical point of view i sto investigate the joint evolution of pairs of variables, pertaining either to the scalar or to th evelocity field, in their associated phase spaces . The phase space for two scalar variables wil ldescribe a dynamical correlation of two somehow connected scalar mixing processes evolvingwith time ; if one of the variables is associated to the scalar field while the other is related t othe velocity field, the result will display the dynamical effect of a turbulence property on thescalar mixing process .

Given pairs of variables, related either to the scalar or to the velocity field, the metho dconsists in conditionally averaging their transport equations to obtain a dynamical system inthe associated 2-D phase space . This idea, introduced by Martín et a1*13* for velocity gradien tinvariants, allows to describe the joint evolution of any pair of variables in a simple, direc tway. The technique can also be useful to isolate the effect of a particular term of the transpor tequation on the global dynamics, enabling the understanding of its role in the correspondin gphysical mechanism .

Given the variables X and Y, the 2-D phase plane XY is considered . Conditional timederivatives defined at each point (X, Y) of that phase space are written a s

X (X, Y)

DX

DtX, Y) (14)

7 3

Y (X Y)D YDt

X, Y)

(15 )

Substituting the values of DX/Dt and DY/Dt one obtains

X(X,Y) = ( T1x X, Y)+(T2xl X,Y)+ . . .

(16)

y(X , Y) = (Tu, X, Y) + (T2Y X, Y) + . . .

(17 )

where T1x, T2x, T1Yi . . ., stand for the different terms on the right hand sides of the transpor t

equations of X and Y. Conditional time derivatives of X and Y, at each point of the plan e

(X, Y), result simply in the summation of the conditional averages of those terms .The vector (X(X,Y), Y(X,Y)) is the velocity at each point of the phase space . Eqs . (16)

and (17) constitute a dynamical system, since that velocity depends solely on the coordinates

(X, Y) . It must be noticed at this point that this dynamical system is, for scalar field relatedvariables, time dependent, since their statistics evolves with time . This fact raises the peed offurther investigation on how the resulting scalar dynamics changes with time and also suggest sexploring the self-similarity and universal behavior of these systems .

3 .1 Numerical experiments and results .

The data fields used to calculate the averaged dynamical systems are the ones already describe d

in the previous section. The velocity field is forced and a Reynolds number Rea 47 is reached .

The value of the Schmidt number, Sc, is 1 .0 ., and the initial scalar field is a "blob" in the middl e

of the cube with value C = 1 . The results correspond to a time when the scalar pdf has relaxe dto a nearly uniform distribution, with a remnant of the two peaks near the extreme values, 0

and 1 .Each figure presents both the joint PDF and the averaged "velocity" for severa]. 2-D phase

planes . Conditional mean trajectories [141 can be calculated from the vector fields . These tra-jectories, not shown in the present work, represent the most probable evolution of the variables ,for prescribed initial values .

The phase planes have been discretized in square bins, in order to numerically calculate th e

required joint PDFs and conditional averages . The size of the bins has been chosen large enoug hto include a significant number of samples, in order to reach stable statistics, and small enoug hto capture the variations of the averages . The isocontours in the joint PDFs are represented in

log-scale . All the numerical values correspond to the DNS units .Figure 14 displays on the left the dynamical system resulting for the phase plane scalar

fluctuation-diffusion term, (c, D0 2 c) . A stable focus at the origin is apparent, with point smoving clockwise spirally until they reach the center ; it can be observed that points with larg evalues of diffusion change the sign of the scalar fluctuation before it becomes zero . Figure14 shows also the result for the square scalar fluctuation-scalar dissipation rate phase plane ,

(c2 , ea), where points with large initial fluctuations strongly increase a moderately small value

of the dissipation ; the larger the values of the scalar fluctuation and of the dissipation rate ,the larger that increment . On the contrary, for small values of c2 the scalar dissipation is a

monotonically decreasing magnitude .

3 .2 Vorticity and strain effects on mixing .

The effects of vorticity and strain on scalar mixing have been studied considering the phase

planes (Q, c2) and (Q, ca) . The second invariant of the velocity gradient tensor, Q = Qw + Qs,

74

-0 504-0 .3-0 .2-01 Ó O i l 02 03 0*4 0 .5

scalar fluctuation

oo

o .b5

0 .1

0 .15

0 .2

square scalar fluctuatio n

Figure 14: Joint PDF and averaged dynamics in the phase planes (c, DV 2 c) and (c2 , cc )

provides a quantitative estimation of the local balance between rotation (Qw = w2/4) and strai n(Qs = -SiiSii/2) . The result for the plane (Q, c 2), shown in the first graphic of Figure 15 ,indicates that the square scalar fluctuations decrease for all values of Q, as it should ; however ,this process is observed to be more intense in the left semiplane (Q < 0) corresponding to hig hstrain and low vorticity regions in the flow . This behavior is better explained through the resul ton the right of the same figure, where c c is found to increase strongly for large strain value s(Q < 0), while it is progressively reduced as the strain diminishes and the vorticity dominate sthe balance (Q > 0) . Both processes, the increase of the scalar dissipation in the left semiplan eand the reduction in the right one, are enhanced for large values of c c .

Q

o

Figure 15 : Joint PDF and averaged dynamics in the phase planes (Q Oc , cc) and (Q, c2 )

4 Scalar with chemical reaction .

Simulation and modelling of variable density turbulent combustion constitutes a extrernely com -plex problem . Dynamically passive chernical kinetics will be assumed as a first simplification,

7 5

considering low Mach number and constant average pressure . Highly diluted concentrationsand not very exothermic reactions are commonly considered, in order to accomplish the dy-namical passivity constraint . Within this framework, a turbulent reacting scalar is considere din this section, and some results are presented .

Considering the aboye assumptions, the flow equations decouple from the scalar field equa-tion, which is written as

ax = DV2 c +íC

+ ui

w(C) . (18)

where w(C) is the chemical source terco .The squared scalar fluctuations transport equation result i n

DV2c 2 — 2€, . + 2c(w(C)— < w(C) >)at (19)+ u*ax

The scalar variance time evolution for the reactive case depends on the scalar dissipationbut also on the functional form of the chemical term . The scalar gradient and scalar dissipatio nrate equations contains also explicit terms depending on the source .

4 .1 DNS of a simple case .

Numerical simulation of a reactive scalar in forced isotropic turbulence has been conducted forthis work . The chemical source term used is w(C) _ -11 .0C2. The Reynolds number basedon the Taylor microscale is Rea ^ 47. The initial scalar distribution is a double Dirac delt awith peaks at the extreme values, 0 and 1 . The Schmidt number is Sc = 1 .0. The result scorrespond to a time when the velocity field has reached a statistically stationary state and th escalar probability density function (pdf) is almost identical everywhere in physical space . Thescalar average at that time is < C >= 0 .063

The contributions of the four velocity field topologies to integrated variables in the flo whave been calculated, similarly as it was done for the inert case in section 2 . The result for th estrain, vorticity, scalar variance, dissipation and dissipation production term, is shown in Tabl e

3. Comparing the results with Table 2, each topology contains for the reactive case simila rfractions of scalar variance, scalar dissipation or dissipation production that those obtained i nthe inert case. So the particular term source chosen here does not introduce changes in th ebehavior of the scalar evolution for the different topologies .

SF/SD>0,R<0

UF/CD>0,R>0

UN/S/SD<0,R>0

SN/S/ SD<0,R< 0

% samples 39.6 28.8 24.5 7 . 2

—Qs 33.9 23.1 36.2 6 . 7Qw 60.6 24 .7 11 .6 3 . 1c 2 41 .16 29 .7 22.0 7 . 1e, 35.9 21 .9 34.6 7 . 5

—2DSi;c, i c,* 23.3 16 .8 53.1 6 . 8

Table 3. Contribution (in percentage) of each topology to the integrated magnitude in the flow .Reactive case .

The effects of the chemical reaction in the scalar diffusion and scalar dissipation are exploredcalculating averaged dynamical systems, as in section 3, for the reactive case . Figure 16 show s

7 6

the resulting picture for the planes (c, D0 2c) and (c2 , Ec ) . Scalar diffusion behaves similarl yto the inert case, producing an stable focus at the origin . On the contrary, chemical reactio ndeeply changes the time evolution of the scalar dissipation, which now decreases in at all point sover the plane, even for the largest scalar fluctuations ; the negative chemical reaction ter mseems to dominate over the production .

Figure 16 : Joint PDF and averaged dynamics in the phase planes (c, DV 2 c) and (c 2 , ce) for areactive scalar with source term w(C) = -11 .0C2 .

5 Conclusions .

Scalar mixing depends on the topologies of the velocity field . This dependence appears to b esignificant only for large values of the velocity gradient invariants, representative of small scal eorganized structures . This indicates that small scale intermittency plays a role in turbulen tmixing mechanisms . The scalar variance appears weakly correlated with velocity field proper-ties, but the scalar dissipation rate presents strong dependence on the strain, and reaches it shighest values at points with UN/S/S topology, which correspond to planar strained struc-tures. Regions of the flow with that topology supply the largest contribution to the total scala rdissipation (i .e., the scalar dissipation integrated over the whole domain) . Averaged produc-tion of scalar dissipation results positive, and presents the largest positive values in UN/S/Stopologies . This is explained from the fact that this kind of topology produces simultaneousl ylarge values of the strain and large values of the scalar dissipation . Small scale topology effect son the scalar field evolution appears to be the same for both the inert and the reactive cases .

The averaged pairwise dynamics of scalar (or velocity) related variables is obtained fromtheir transport equations using conditional averages . This method allows to isolate the effec tof different mechanisms as diffusion, dissipation or chemical reaction on scalar evolution . Theinfluence of the local topology motion along the scalar evolution can be explicitly describe dand visualized .

Results presented in this work correspond to an interrnediate time along the scalar fiel devolution, and also to a particular Schmidt number (1 .0) . Much work is needed to investigat edifferent cases considering several stages of the scalar field evolution, different Schmidt number sand also different chemical source terms .

Chem . reaction w(C)=-11'C' C

414

-0 .05 ' ó 'óá ''-ó 'ós '-o 'o2 '-ó"b

'ó_bi' o.bz '

scalar fluctuation

ti '44

0 .0300 ' o'obóy' o :ob02 o'obós' 'o :obó4 'o'obo ssquared scalar fluctuation

7 7

Acknowledgements .

This work has been partially supported by the Spanish Ministry of Science and Technology i nthe frame of the research project BFM2000-1065 .

References

[1] Blasco, J .A., Fueyo, N., Dopazo, C . and Ballester, J . 1998 . "Modelling the temporal evo-lution of a reduced combustion chemical system with an artificial neural network", Com-bustion and Flame 113, 38-52 .

[2] Fueyo, N ., Vicente, W ., Blasco, J . and Dopazo, C . 2000. "Stochastic simulation of NOformation in lean premixed methane fiames", Combustion Science and Technology, 150,

1-17 .

[3] Tsinober, A . 2001 . "An informal introduction to turbulence " , Kluwer Academic Publ . TheNetherlands .

[4] Kerr, R .M. 1985. "Higher-order derivative correlations and the alignment of small-scal estructures in isotropic numerical turbulence" , J . Fluid Mech . 153,31 .

[5] Ashurst, W.T., Kerstein, A.R., Kerr, R.M., and Gibson, C.H. 1987. "Alignment of vorticit yand scalar gradient with strain rate in simulated Navier-Stokes turbulence",Phys . Fluid sA, 8(30), 2343-2353 .

Ruestch, G .R. and Maxey, M.R. 1991 . "Small-scale features of vorticity and passive scalarfields in homogeneous isotropic turbulence", Phys. Fluids A, 3(6), 1587-1597 .

Pumir, A. 1994 . "A numerical study of the mixing of a passive scalar in three dimension sin the presence of a mean gradient" Phys. Fluids, 6(6), 2118-2132 .

Ottino, J . M . 1989. "The Kinematics of Mixing : Stretching, Chaos and Transport", Cam-bridge Univ . Press, N.Y.

Dopazo, C . 1994. In Turbulent reacting flows, Eds. Libby, P.A. and Williams, F .A., Ch 7 ,375, Academic Press .

[10] Chong, M .S., Perry, A .E., and Cantwell, B.J . 1990 . "A general classification of three-dimensional flow fields", Phys. Fluids A, 2, 5, 765-777 .

[11] Soria, J ., Chong, M .S., Sondergaard, R., Perry, A.E. and Cantwell, B .J. 1994. "A study ofthe fine scale motions of incompressible time-developing mixing layers", Phys . Fluids A,6, 871-884 .

[12] Jiménez, J .,Wray, A ., Saffman, P . and Rogallo, R . 1993 . "The structure of intense vorticityin homogeneous turbulence " , J . Fluid Mech ., 255, 65 .

[13] Martín, J ., Ooi, A., Chong, M.S . and Soria, J . 1998 . "Dynamics of the velocity gradienttensar invariants in isotropic turbulence " , Phys. Fluids, 10(9), 2336-2346 .

[14] Ooi, A., Martín, J ., Soria, J . and Chong, M .S. 1999, "A study of the evolution and charac-teristics of the invariants of the velocity gradient tensor in isotropic turbulence", J . FluidMech ., 382, 141-174 .

[6 ]

[7]

[8 ]

[ 9 ]

7 8

[15] Chen, J .H., Chong, M .S., Soria, J ., Sondergaard, R., Perry, A.E., Rogers,M ., Moser, R . andCantwell, B .J . 1990 . "A study of the topology of dissipating motions in direct numericalsimulations of time-developing compressible and incompressible mixing layer s" , Proceedingsof the 1990 CTR Summer Program, Standford, CA .

[16] Cantwell, B .J . 1993. "On the behavior of velocity gradient tensor invariants in direc tnumerical simulations of turbulence " , Phys. Fluids A, 5, 2008-2013.

[17] Blackburn, H.M ., Mansour, N .N . and Cantwell, B .J. 1996 Topology of fine scale motionsin turbulent channel fiow . J. Fluid Mech ., 310, 269-292 .