Modeling of turbulence-chemical reaction interactiocontinues to present a challenging task for engineersscientists and remains an active area of research.1–5 Whilethere has been significant progress in understandingmodeling of turbulence and chemical reaction separatmuch less is known about their coupled behavior.6 The non-linear interactions between turbulence and chemical reacoccur over a wide range of time and length scales andvolve many different quantities. Our lack of adequate undstanding of these interactions imposes serious limitationsthe modeling of chemically reactive turbulent flows. For eample, the majority of existent turbulence closures whichused for reacting flow calculations are based on those deoped for nonreacting flows. These closures are potentilimited and cannot account for important phenomena inbulent combustion such as the extensive density and mollar property variations, significant dilatational turbulent mtions, etc.
Previous numerical and experimental investigationsvolving flame-turbulence interactions primarily discuss tinfluence of the coherent structures on mixing and reactiofree shear flows7–13 ~for recent reviews see Givi,1
a!Present address: Department of Mechanical and Nuclear Engineering,sas State University, Manhattan, KS 66506-5205.
b!Author to whom correspondence should be addressed; [email protected]
1181070-6631/2000/12(5)/1189/21/$17.00
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Dimotakis,14 Drummond and Givi,15 and Coats16!. The re-sults of these investigations indicate the importance oflarge-scale as well as small-scale structures on the mixand reaction. They also show that these structures are sigcantly affected by the exothermicity of the reaction andflow compressibility. The effects of two-dimensional anthree-dimensional turbulence on the structure of premiand diffusion flames in shearless mixing layer are studiedHaworth and Poinsot,17 Mahalingamet al.,18 and Boratavet al.19 via direct numerical simulation~DNS!. Their resultsindicate that the structure of the flame is significantly alteby the turbulence.
Mixing and reaction in homogeneous turbulence haalso been the subject of numerous investigations.20–33 In themajority of these investigations the flow is considered toincompressible and the scalars are considered to be pasLeonard and Hill,25 and Nomura and Elgobashi,28 study thestructure of the reaction zone in homogeneous isotropicshear incompressible turbulence via DNS. Their results shthat the intense reaction rates occur over the regions wthe concentration gradient is large and the gradients ofscalars tend to align with the direction of the most comprsive eigenvector of the strain rate tensor. The effect ofcompressibility on the turbulent mixing in homogeneoshear turbulence is studied by Blaisdellet al.34 and Caiet al.35 Their DNS results indicate that the dilatational covective velocity does not contribute noticeably to the mixiof the scalars.
1190 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
The effects of the chemical heat release on the isotrodecaying and forced turbulence were recently studied by Bakrishnan et al.,36 Jaberi and Madnia,37 and Martin andCandler,38 who consider different initial scalar conditionand chemical kinetics. Although these investigations revsome interesting features of the turbulence-chemical reacinteractions in isotropic turbulence, more detailed studhave to be performed in order to fully understand the coplex role portrayed by the combined influences of the turlence and the chemical reaction in a compressible fluiddium. In this study, we use the data gathered from DNSisotropic decaying turbulent reacting flows to further eludate the interactions between turbulence and chemical rtion. The main objectives of this investigation are:~1! toanalyze the flow and the flame structure,~2! to study theeffects of the heat of reaction on different modes of enerand ~3! to examine the dynamical evolution of the vorticand the dilatational fluid motions and their correlation in tpresence of chemical reaction in turbulent flows.
This paper is organized as follows. In Sec. II the goerning equations are presented and the computational modology for solving these equations is explained. The respertaining to flame characteristics and the effect of heareaction on the flow structure, turbulent energy, and thelenoidal and dilatational turbulent motions are presentedSec. III. A summary of important findings and conclusionsgiven in Sec. IV.
II. GOVERNING EQUATIONS AND COMPUTATIONALMETHODOLOGY
The primary independent transport variables in a copressible flow undergoing chemical reaction are the denr, the velocity components inxi direction ui , the specificenergye, the pressurep, the temperatureT, and the massfraction of speciesa,Ya . The conservation equations goerning these variables are expressed as
]r
]t1
]
]xj~ruj !50, ~1!
]
]t~rui !1
]
]xj~ruiuj !52
]p
]xi1
1
Reo
]Q i j
]xj, ~2!
]
]t~re!1
]
]xj~ruje!52
]
]xj~puj !
11
~g21!Mo2 Reo Pr
]
]xjS m
]T
]xjD
11
Reo
]
]xj~uiQ i j !1Q, ~3!
]
]t~rYa!1
]
]xj~rujYa!5
1
Reo Sc
]
]xjS m
]Ya
]xjD1wa ,
~4!
where
Q i j 52mSi j 22
3mDd i j ,
icl-
alons--
e-f
-c-
y,
-th-tsf
o-in
-ty
Si j 5(1/2)(]ui /]xj1]uj /]xi) is the strain rate tensor,D5]uk /]xk is the dilatation,d i j is the Kronecker delta, andwa andQ represent the chemical mass and heat source terespectively. The nondimensional viscosity,m, is modeled as
m5Tn. ~5!
The specific energy is the summation of the specific inter(eI) and kinetic (eK) energies
e5eI1eK5p
r~g21!1
1
2uiui , ~6!
and thermodynamic variables are related through the eqtion of state
p5rT
gMo2 . ~7!
All variables in the above equations are normalized usreference length (l 0), velocity (u0), temperature (T0), anddensity (r0) scales. Consequently, the important nondimesional parameters are the box Reynolds number,o5rouolo /mo , the Prandtl number, Pr5mocp /ko , the Schmidtnumber, Sc5mo /roDo and the reference Mach numbeMo5uo /AgRTo ~R is the gas constant!. The viscosity,mo ,thermal diffusivity, ko , and mass diffusivity,Do , are as-sumed to be proportional toTo
n , the specific heat at constanpressure,cp , is constant and in all cases the Lewis numbeunity with Pr5Sc50.7. Also, in all simulationsMo50.6 andg51.4. The gas is assumed to be calorically perfect.
The chemistry is modeled with a single-step irreversireaction of the typeA1rB→(11r )P ~r 51 in this study!with an Arrhenius reaction rate,
wA5wB521
2wP52Dar2YAYB exp~2Ze/T!,
Q5Ce
~g21!Mo2 wP . ~8!
The mass fractions and the reaction rates of speciesA, B, Pare represented byYA ,YB ,YP andwA ,wB ,wP , respectively.The mass fraction of the mixture fraction,Z, is representedby YZ . All the species are assumed to be thermodynamicidentical. The constant nondimensional quantities affectthe chemistry are the heat release parameter52H0/cpTo , the Damko¨hler number Da5K frol o /uo , andthe Zeldovich number Ze5Ea /RTo , where2H0 is the heatof reaction,K f is the reaction rate parameter~assumed to beconstant!, andEa is the activation energy.
Equations~1!–~4! are integrated using the Fourier psedospectral method39,40 with triply periodic boundary condi-tions. The explicit second-order accurate Adams–Bashfoscheme is used to time advance the variables. All simulatiare conducted within a box containing 1283 collocationpoints. Aliasing errors are treated by truncating the Fouvalues outside the shell with wave numberkmax5&N/3~whereN is the number of grid points in each direction!. Thevelocity field is initialized as a random solenoidal, thredimensional field with a zero mean and Gaussian specdensity function. The initial degree of compressibility is co
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1191Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
trolled by varying the ratio of the energy residing in dilattional motions to the total energy.41 The initial pressure fluc-tuations are evaluated from a Poisson equation. The indensity field has unity mean value and no turbulent flucttions and the initial values of the temperature are calculafrom Eq. ~7!. The initial velocity field is allowed to decay toa ‘‘self-similar’’ state, corresponding to a fully developeturbulent flow, before scalar mixing and reaction begin. Tcorresponds to timet50 on all the figures. The scalarA isspecified in the physical domain in such a way as to yielsquare wave in thex2-direction ~slabs!.32 The slabs are approximated by an error function distribution such that tscalar field varies smoothly in the range 0,YA,1. The sca-lar values are constant inx12x3 planes along thex2-direction. Cases with four scalar slabs are considered.initial probability distribution function~PDF! of the scalarAis approximately composed of two delta functions centeat YA50,1. The scalarB is perfectly anticorrelated withAand there are no productsP in the domain at initial time.
III. RESULTS AND DISCUSSIONS
Direct numerical simulations of chemically reacting istropic compressible decaying turbulence are performTable I provides the listing of the relevant information aboeach of the cases studied. The variablen in this table denotesthe temperature exponent in Eq.~5!. Cases 1 and 2 are threference cases in which the reaction is constant rate witheat release. The magnitude ofr CL ~the ratio of the dilata-tional kinetic energy to the total kinetic energy as definbelow! is different in these two cases. The initial valuer CL is very small in case 1 but is significant in case 2. Ca3–9 are considered to investigate the effects of heat reldue to chemical reaction on the velocity, pressure, tempture, density, and scalar statistics. In cases 2 and 9 the inflow compressibility is higher than the other cases listedTable I. Case 4 is similar to case 3 but with constant molelar viscosity, conductivity, and diffusivity coefficients. Thcase is considered in order to isolate the effects of healease on the molecular transport properties. Case 5 issimilar to case 3 but in this case in addition to Eqs.~1!–~4!an extra energy equation is solved in which the heat releterm is neglected. The temperature obtained from this ational equation is used to calculate the pressure and thesity. Therefore, in case 5 the effect of heat release on
aIn this special case the effect of chemical reaction on turbulence ismoved.
al-d
s
a
he
d
d.t
no
d
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velocity field is eliminated. Case 5 is considered in orderisolate the effects of heat release on the mixing and reactThe initial velocity field in case 1 and cases 3–8 are identwith Reynolds number based on Taylor microscale, Rl
550.1 ~Rel550.7 for cases 4 and 5 due to the differencemolecular viscosity! and teddy51.8 ~teddy is the large scaleeddy turnover time! and exhibits almost no contribution fromthe dilatational fluid motions. In cases 2 and 9, Rel553.7and teddy51.92 at the initial time.
The temporal evolutions of the statistical quantitiesextracted from DNS are of primary importance here. Thestatistics are obtained by volumetric averaging over allcollocation points. Two of the important statistical quantitiare the mean and the variance of a variablea which aredenoted by^a& and sa5^(a2^a&)2&, respectively. Thevariance of a vector is defined as the average of the variaof its three components. The three-dimensional~3D! spectraldensity function of the variablea ~a[v,T,p,r,A,Z etc.! isidentified byEa(k). The other important quantities are thlocal turbulent Mach number,M, and the enstrophy,V,which are defined as
M5Auiui
AgRT, V5
1
2v iv i , ~9!
wherev i denotes the vorticity vector. Of particular intereare the statistics of the solenoidal and the dilatational coponents of the velocity and the kinetic energy. To obtathese statistics, first the velocity~or the kinetic energy! isdecomposed into the solenoidal and the dilatational partscording to the Weyl decomposition.41–43 Then the statisticsare calculated for each component separately. In the dission of the results below, the superscripts ‘‘s’’ and ‘‘ d’’ de-note the statistics that are calculated from the solenoidalthe dilatational velocity~or kinetic energy! components, re-spectively. For example,Ev
s(k) and Evd(k) denote the spec
tral density functions of the solenoidal and the dilatationvelocities, respectively.
In the presentation of the results below, the ratios
r CS5^D2&
^V&1^D2&, r CL5
svd
svs1sv
d , ~10!
represent the flow compressibility at small~dissipation! andlarge~energy containing! scales, respectively.44 The correla-tion coefficient between two variablesa and b, z(a,b) isdefined as
z~a,b![^ab&2^a&^b&
@~^a2&2^a&^a&!~^b2&2^b&^b&!#1/2. ~11!
A. Flame characteristics
Characteristics of the flames considered in this studyidentified via analysis of the flame structure. In turbulereacting flows, the spatial and the compositional structurethe flame is dependent on the flow~turbulence! structure aswell as the chemistry parameters. The flame structure isdependent on the variations of the thermodynamic quantsince the reaction rate is dependent on these quantities.
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1192 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
heat of reaction, in turn, affects the turbulent motions andthermodynamic variables, hence a two-way coupling exis
In the assessment of the reaction, it is useful to studyreaction rate (w[2wA) and its components; the mixinterm (G5r2YAYB) and the temperature-dependent term@F5exp(2Ze/T)#. In the nonheat-releasing case 1, the reactis solely dependent onG and its mean values as shownFig. 1~a!, after reaching a peak att'2, decay continuouslyHowever, in the heat-releasing cases, the reaction ratepends on bothG andF and its maximum mean values occat different times for different values of Da, Ze, and CExpectedly,^w& peaks earlier as the values of Da andincrease or those of Ze decrease. The mean reaction ranot, however, very much dependent on the initial flow copressibility as the results corresponding to cases 6 and 9very close. A comparison between the results in cases 34 also indicates that the variation in the magnitudes ofmolecular transport coefficients has little effect on the teporal variation of^w&. However, the Reynolds numberstrongly affected by the variation in the molecular viscosiThis is illustrated in Fig. 1~b!, where the time evolution oRel is presented. The initial values of Rel vary between 50.1
FIG. 1. Temporal variation of~a! the mean reaction rate,~b! Rel , and ~c!the mean flame surface density.
e.e
n
e-
.
is-re
nde-
.
~cases 1, 3, 6, 7, and 8! and 53.7~cases 2 and 9!. At t52.5, when the mean reaction rates for cases 3 and 4 rtheir peak values, the values of Rel are between 17.5~case 3!and 36.4~case 4!. The Reynolds number in case 4 is highthan that in case 1 due to the amplification of the dilatatiofluid motions by the heat of reaction.
In modeling of turbulent reactive flows, the ‘‘flameleassumption’’ is usually invoked. With this assumption, treaction rate can be directly related to the mixture fractthrough the flame surface density,S.5,45,46In turbulent com-bustion, the flame surface density is a complex functionthe flow and chemistry parameters. Following Pope,47 wedefine the average flame surface density as
^S&5^u¹YZud@YZ2~YZ!st#&
5^u¹YZuuYZ5~YZ!st&
3P~YZ5~YZ!st!, ~12!
where d is the Dirac-delta function,quYZ5(YZ)st& is theexpected value of the quantityq conditioned on YZ
5(YZ)st , and P(YZ5(YZ)st) denotes the probability thaYZ5(YZ)st . Figure 1~c! shows the temporal evolution of thmean flame surface density for different cases. At eatimes, the values ofS& are relatively low and similar in allcases due to initial conditions. However, the flame surfdensity increases due to turbulent stretching/folding, andter peaking at 2,t,3, decays due to scalar mixing and tubulence decay. A comparison among the results for casand 6 indicates that the late time values of^S& are lower forhigher values of the heat release parameter. This is primadue to variation of the molecular transport coefficients wtemperature as the results in the heat-releasing case 4constant molecular coefficients are similar to those in nheat-releasing case 1. This observation is further suppoby the results obtained for case 7, which has a higher vaof Da than case 6 and exhibits higher temperatures at eatimes. The small-scale scalar fluctuations are dependenthe magnitudes of the molecular transport coefficientsdecay faster at higher temperatures, which results in lovalues ofS. However, it should be noted that the averau¹YZu conditioned on (YZ)st and probability of (YZ)st ,which appear in the flame surface density definition@Eq.~12!#, are both affected by the heat release~not shown here!.For case 4, these terms counteract each other and theeffect on the flame surface density is small.
A comparison between Figs. 1~a! and 1~c! clearly indi-cates that the temporal evolution of the mean flame surfdensity is very different from that of the mean reaction raand the finite rate chemistry effects are important. Hewand Madnia,48 and Pierce and Moin,49 also found that thefinite rate chemistry effects become important under cerreacting flow conditions. To further explain the resultsFig. 1, the temporal variation of the volumetric averagvalues of ln(G), ln(F), and ln(w) for case 3 are considered iFig. 2. The results for other heat-releasing cases are quatively similar. Figure 2 shows that the evolution of the mereaction rate is different than the mixing and thtemperature-dependent terms. Expectedly, the mean va
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1193Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
of the temperature-dependent term,F, are very low at earlytimes but increase significantly by the heat of reaction. Tmixing term,G, exhibits behavior similar to that shown fothe flame surface density in Fig. 1~c!. The volumetric aver-aged values of this term increase at early times due to mixof the reactants and decrease later due to consumption oreactants. Att52.5, when the mean reaction rate peaks~Fig.1!, the average values ofF andG are comparable, showinthe important contribution of both mixing and temperaturethe reaction rate.
In order to further identify the role of reaction in thflow field, three regions are defined based on the meanues ofw, F, andG at t52.5. Table II presents the percentaof the computational grid points corresponding to these thregions,
~i! Region I:w.^w& t52.5,~ii ! Region II: w,^w& t52.5,G.^G& t52.5,F,^F& t52.5,~iii ! Region III: w,^w& t52.5,G,^G& t52.5,F.^F& t52.5,
at different times. The regions I, II, and III may be associawith the ‘‘reaction region,’’ the ‘‘mixing region,’’ and the‘‘hot-product region,’’ respectively. The results in Tableare consistent with the results shown in Fig. 6 below, aindicate that att51 a significant portion of the domain consists of region II. At this time the reaction rate and the teperature are relatively low and regions I and III have a nligible contribution. At t52.5, when the mean reaction rais significant, most of the field is composed of region I.later times (t54), when the mean reaction rate is small, tmixing and the reaction occur rarely and the total voluoccupied by regions I and II is less than 1% of the comtational domain.
FIG. 2. Time variation of different terms in the reaction rate expression@Eq.~8!# for case 3.
TABLE II. Percentage of the domain filled with regions I, II, and III.
Region I Region II Region III
t51 0% 75% 0%t52.5 40% 15% 26%t54 0.5% 0% 93.5%
e
gthe
l-
e
d
d
--
e-
The results in Figs. 1 and 2 and Table II show the avage behavior of the reaction at different times and doprovide information about the flame structure. The flamstructure can be studied by examination of the reactive scies in mixture fraction~compositional! space. Figure 3shows the scatter plots of the reactants and the product mfractions in mixture fraction space for case 3. These scaplots are similar to those of a typical flame with finite rachemistry effects.
The scatter plot of the reaction rate in mixture fractispace for case 3 att52.5 is shown in Fig. 4~a!. This figureindicates that the conditional mean value of the reactionis the highest at the stoichiometric surface (YZ50.5), similarto that observed in Fig. 3 for product mass fraction. Hoever, the variation of the reaction rate in mixture fractispace is different than that of the product mass fraction.explain this behavior, the scatter plots ofG andF parts of thereaction rate are shown in Figs. 4~b! and 4~c!. TheF term isonly a function of temperature and behaves similar toYP .The behavior of theG term is more complex. This termdepends on the density and the product of the reactants’ mfractions, and as shown in Fig. 4~c! attains relatively lowvalues near the stoichiometric surface, mainly due to lvalues of the density@Fig. 4~d!#. At later times (t.2.5), asshown in Fig. 2, theF term is larger than theG term and thescatter plots ofw exhibit behavior similar toF and YP . Itshould be noted that, due to homogeneity of the flow,reaction in this study occurs through a constant volume pcess. Therefore, the mean density is constant but the mpressure and temperature increase continuously and slarly. Within the reaction zone, the density decreasesto volumetric flow expansion and is lower than unity. Ouside this zone the density is higher than unity, as shownFig. 4~d!.
It is shown above that in exothermic reacting casesmixing term and the temperature-dependent term both ctribute significantly to the reaction. However, the mean vues of these terms evolve very differently. This raisesquestion of how the local values of these terms are correlawith each other and with the reaction rate. To answer tquestion, the temporal variation of the correlation coefficie
FIG. 3. Scatter plots of the reactants and product mass fractions in mixfraction ~Z! space for case 3 att52.5 ~sample size 4096 points!. The solidlines represent the conditional means and are calculated based on alpoints (1283).
alnts
1194 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
FIG. 4. Scatter plots of~a! w, ~b! F, ~c! G, and~d! r, inmixture fraction space for case 3 att52.5 ~sample size4096 points!. The solid lines represent the conditionmeans and are calculated based on all grid poi(1283).
ahetheia
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ith-
ith-
nt
w
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n ind-s
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betweenG and w and that betweenG and F for case 3 areshown in Fig. 5. The results for other heat-releasing casesqualitatively similar and are not shown. It is evident that tG term is not well correlated with the reaction rate whenrate of reaction is significant. This lack of correlation btween G and w has a significant influence on the spatflame structure and is explained below.
Figure 6 shows the joint PDF ofG andF and the jointPDF of G and w at several different times. At early time(t51), as indicated in Table II, in most of the domain treaction is insignificant and is confined to the mixing zonThis explains the strong correlation betweenw andG at earlytimes as observed in Fig. 5. Also, at early times the tempture increases only in the regions where the reaction taplace and there is a good correlation betweenF andG ~Fig.5!. The joint PDF plot in Fig. 6~b! confirms thatF andG areindeed well correlated. Att52.5, when the mean reactio
FIG. 5. Temporal variation of the correlation coefficient betweenG andFandG andw, in case 3.
re
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.
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rate is high, the field is primarily composed of zones whigh temperature~hot-product region, 26% of the total number of points, and reaction region withG,^G&,22% of thefield! and of zones with mixed reactants~mixing region, 15%of the field, and low-temperature reaction region wF,^F&,10%!. In the zones with mixed reactants the temperature has not yet increased significantly, soG.^G& andF,^F& in these zones@Fig. 6~d!#. In the high-temperaturezones,G,^G& and F.^F&. These results are consistewith Fig. 5, where it is shown thatF and G are negativelycorrelated att52.5. Although there are regions in the flowith very high values ofG andw, in most of the domain thevalues ofG andw are moderate to low@Fig. 6~c!#. The pointsin the domain are relatively evenly distributed among tfour regions defined by the linesG5^G& and w5^w& onFig. 6~c!. As a result, the correlation betweenG and thereaction rate is very low att52.5 ~Fig. 5!. At later times,most of the reactants have already burnt and, as showTable II, the field is occupied almost entirely by hot proucts. At these times,F is almost evenly distributed around itmean@Fig. 6~f!# and the correlation betweenF andG is low.At elevated temperatures, the exponential term has smspatial variation andF is nearly constant. Therefore, the raction rate as illustrated in Figs. 5 and 6~e! is highly corre-lated with G. Figure 6~e! also shows that att54 there arestill some rare regions in which mixing and reaction are bsignificant.
So far, we have only discussed the influence of varioparameters on the reaction. In the following sections thefects of reaction on turbulence are studied. In particularwill show how the turbulence structure, the turbulent enerthe vorticity field, and the dilatation field are affected by treaction. The influence of the reaction on the thermodyna
1195Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
FIG. 6. Joint PDF of~a! w andG at t51, ~b! G andF at t51, ~c! w andG at t52.5, ~d! G andF at t52.5, ~e! w andG at t54, ~f! G andF at t54, for case3. The thick lines represent the average values which are respectively:^G& t51.050.087, ^w& t51.050.017, ^F& t51.050.0008, ^G& t52.550.017, ^w& t52.5
variables is also studied. The results are consistent with thof Balakrishnanet al.,36 Jaberi and Madnia,37 and Martin andCandler,38 and indicate that the fluctuations of the tempeture and pressure increase significantly by the heat of r
se
-c-
tion. Although the value of density is minimum at the stichiometric surface@Fig. 4~d!#, its fluctuations increase as thrate of heat release increases. The rate of heat releadependent on the density, temperature, and reactants’ m
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1196 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
fractions and varies significantly throughout the flow fieThe nonuniform generation of heat results in the enhanment of the fluctuations of thermodynamic variables. Tvariations in temperature and density in turn result in mofication of the reaction rate.
B. Flow structure
The heat of reaction has a noticeable influence on turlence structures. This is partially demonstrated in this secvia analysis of the strain rate tensor@Si j 51/2(]ui /]xj
1]uj /]xi)# and alignment of its eigenvectors with the voticity and scalar gradient vectors. The eigenvalues ofstrain tensor or the principal strain rates are termed convtionally asa, b, g, with a.b.g. The PDFs of the eigenvalues of the strain rate normalized by the magnitude of tstrain (ueu5(a21b21g2)1/2) for cases 1 and 3 att52.5 areshown in Fig. 7. It is observed that although the shape ofPDFs is not substantially affected, the heat of reactioncreases the variances of all eigenvalues. In agreementthe previous observations,22,25,50the values ofb are shown tobe mostly positive. Also, due to homogeneity in all casconsidered in this studya1b1g&'0. The reaction hasalso a noticeable influence on the average values of the pcipal strain rates. This is observed in Fig. 8, where itshown that the magnitudes of^a&, ^b&, and ^g& decay faster
FIG. 7. PDF of the normalized eigenvalues of the strain rate tensor for c1 and 3 att52.5.
.e-ei-
u-n
en-
al
e-ith
s
in-s
as the heat release increases. The heat of reaction increthe magnitudes of the molecular transport coefficientsresults in faster decay of the turbulence and the strain eigvalues.
To further examine the effect of reaction on the flostructures, the PDFs of the cosines of angles betweenvorticity and the principal strain directions for cases 1 andare plotted in Fig. 9~a!. The angles between the vorticitvector anda-, b-, andg-eigenvectors are denoted byz1 , z2 ,andz3 , respectively. Direct numerical simulations of incompressible nonreacting turbulent flows suggest a preferenalignment among the strain eigenvectors, the vorticity vecand the scalar gradient vector.22,28,51–55 This preferentialalignment is due mainly to local effects associated withstructure and dynamics of vorticity vector and strain ratensor,26,56–58although the formation of distinct spatial strutures can also affect the alignment.58 It has been found thathe vorticity vector tends to align with the direction of thintermediate~b! strain eigenvector and the scalar gradievector tends to align with the direction of the most comprsive ~g! strain eigenvector. The PDF plots in Fig. 9~a! alsoshow that the vorticity vector tends to be parallel to tb-eigenvector and perpendicular to theg-eigenvector in bothheat releasing and non-heat-releasing cases. The effeheat of reaction is to decrease the alignment of the vortivector with theb-eigenvector and to increase the alignmewith the a-eigenvector. It is interesting to note that theresults, as obtained for a homogeneous~constant volume!flow, agree qualitatively with the results of Nomura anElgobashi59 obtained for an inhomogeneous flow with costant molecular coefficients and infinitely fast chemistry. TPDFs obtained for case 4~not shown here! are similar tothose shown in Fig. 9~a! for case 1. This suggests that thmodification of the alignment of the vorticity vector anstrain eigenvectors is primarily due to variation of the mlecular transport coefficients with temperature.
A more detailed analysis of our results indicates thatalignment between the vorticity vector and the strain eigvectors is more significantly affected by the reaction in t‘‘reaction zones,’’ where the values of the mixture fractioare close to the stoichiometric values. Our results are aconsistent with those obtained by Boratavet al.,19 and indi-cate that in the reaction zones the flow is dominated by st
es
e
FIG. 8. Temporal variation of the mean values of thstrain rate eigenvalues.
ofns-
icate
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1197Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
rather than rotation. This is demonstrated in Fig. 9~b!, wherethe PDFs ofC, sampled over three different ranges ofYZ ,are considered. The strain-enstrophy angle,C, is defined as19
C5tan21Si j Si j
Ri j Ri j,
whereRi j is the rotation tensor. By definition, large valuesC ~@45°! correspond to the strain-dominated flow regioand small values ofC ~!45°! correspond to the enstrophydominated regions. Figure 9~b! shows that in the reactionzones the flow is dominated by the strain and the vortfluid motions play a lesser role. The conditional expecvalues of the enstrophy, conditioned onYZ , for the heat-releasing cases~not shown! are also consistent with Fig. 9~b!
FIG. 9. ~a! PDFs of the cosine of the angles between the eigenvectors ostrain rate tensor and the vorticity vector att52.5, ~b! PDFs ofC for case3 at t52.5, ~c! PDF of the cosine of the angles between the eigenvectorthe strain rate tensor and the gradient ofYZ .
ld
and indicate that the lowest and highest values are obtainside and outside the reaction zone, respectively.
Figure 9~c! shows the PDF of the cosine of the angbetween the scalar gradient vector and the principal stdirections for cases 1 and 3 att52.5. The angles witha-, b-,andg-eigenvectors are denoted byx1 , x2 , andx3 , respec-tively. In both cases, the scalar gradient tends to align wthe most compressive principal direction of the strain rtensor as observed previously.50,54,55Our results~not shown!also indicate that in both heat-releasing and non-hereleasing cases the scalar gradient tends to be mainly noto the vorticity vector. The heat of reaction has little effeon the alignment of strain eigenvectors and the scalar grent. The main effect of the reaction is that the PDFscos(x1) and cos(x2) are closer to each other in heat-releasicases. A comparison with the results obtained for case 4~notshown here! again suggests that the change in the alignmamong strain eigenvectors and the scalar gradient is duvariation of the molecular transport coefficients with temperature.
C. Turbulent energy
In this section the effects of reaction on the turbulekinetic energy and the energy transfer among different coponents of the kinetic energy and the internal energystudied. The effect of heat of reaction on the mean turbukinetic energy is shown in Fig. 10~a!. This figure shows thatthe reaction has little effect on the decay of the turbulkinetic energy~compare cases 1 and 3!. The reason that thedecay rate of the kinetic energy is not significantly affectby the reaction is explained by considering the effectsreaction on different components of the kinetic energy. Tturbulent kinetic energy is composed of the rotational~sole-noidal! and the compressive~dilatational! components. In theabsence of heat release, the values of the dilatational andsolenoidal turbulent kinetic energies decay slowly due to vcous dissipation. However, both components of the kineenergy are affected by the reaction@Figs. 10~b! and 10~c!#.The results in Fig. 10~b! indicate that the solenoidal kinetienergy decays faster due to the heat of reaction~comparecases 1 and 3!. This is primarily due to an increase in thmagnitudes of the molecular transport coefficients and turlent kinetic energy dissipation with temperature. The inflence of the heat of reaction on the dilatational componenthe kinetic energy is different than that on the solenoicomponent. Figure 10~c! shows that the mean values of thdilatational kinetic energy increase significantly due to hrelease, despite the fact that the magnitude of turbulent Mnumber decreases by the reaction. Figure 10~c! also showsthat the generated dilatational motions remain significalong after the reaction is completed. The results correspoing to cases 3 and 4 indicate that the variation in the magtudes of the molecular diffusivity coefficients does not haa significant effect on the evolution of dilatational kinetenergy.
Figures 10~b! and 10~c!, therefore, explain the resultshown in Fig. 10~a!. The solenoidal kinetic energy is affecte
he
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1198 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
by the reaction, primarily due to variation in molecular cefficients. In contrast, the reaction increases the values omean dilatational kinetic energy. The net effect on kineenergy would be the summation of the effects on its solendal and dilatational components. A comparison betwecases 1 and 3 in Fig. 10~a! indicates that the reaction slightlmodifies the decay rate of the mean kinetic energy. In casthe molecular viscosity is constant and the mean valuethe solenoidal energy are not significantly affected byreaction. However, the dilatational energy increases substially by the reaction. Consequently, the mean values ofkinetic energy in case 4 are significantly higher than thoscase 1. From the results presented in Fig. 10 it can becluded that the volumetric flow expansion, on average, dnot have a significant effect on the solenoidal turbulent mtions.
Figure 10 shows that the heat of reaction has a signcant influence on the turbulent energy. However, this fig
FIG. 10. Temporal variations of~a! the turbulent kinetic energy,~b! thesolenoidal component of the turbulent kinetic energy, and~c! the dilatationalcomponent of the turbulent kinetic energy.
heci-n
4,ofen-e
inn-s-
-e
cannot reveal how different turbulent scales are affectedthe reaction. The interaction between turbulence and checal reaction occurs over a variety of different length scaand it is important, from both physical understanding anmodeling point of view, to assess the influence of reactiondifferent flow scales. To address this issue, the 3D specdensity function of the solenoidal and the dilatational veloity for several different cases are considered in Fig. 11. Ishown in Fig. 11~a! that the large-scale solenoidal velocifield is not noticeably altered by the reaction and is similarcases with constant and temperature-dependent diffuscoefficients. However, the small-scale solenoidal turbulmotions are dependent on the magnitudes of the molectransport coefficients and are affected by the heat of reactAs compared to case 1, the magnitudes of the moleccoefficients are higher and the small scales decay fastecase 3. The small-scale values of the solenoidal energcase 4 are slightly higher than those in case 1, since in c4 a net energy is transferred from the internal energy tokinetic energy by the pressure-dilatation correlations. Thiexplained in more detail below, where the transport eqtions for internal and kinetic energies are considered.
In contrast to the solenoidal velocity spectrum, the ditational velocity spectrum is significantly affected by thheat release at all length scales. This is demonstrated in11~b!, where it is shown that the large- and the small-scdilatational energy in cases 3 and 4 is significantly highthan that in cases 1 and 5. Nevertheless, the spectra in c3 and 4 are very close to each other. This suggests tha
FIG. 11. Three-dimensional spectral density functions of~a! the solenoidalvelocity, ~b! the dilatational velocity, att53.
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1199Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
dilatational velocity field is modified by the heat of reactioprimarily due to volumetric flow expansion/contraction, athe variation in molecular transport coefficients has littlefect.
The effect of the initial flow compressibility on the decay of the mean turbulent kinetic energy for both non-hereleasing and heat-releasing cases is shown in Fig. 12non-heat-releasing cases, the kinetic energy decays mslowly as the flow compressibility increases. The resultsFig. 12 are also consistent with those in Fig. 10~a! and indi-cate that the decay of the turbulent kinetic energy issignificantly affected by the heat of reaction when the initflow compressibility is small~compare cases 1 and 6!. How-ever, the reaction changes the mean kinetic energy, wheninitial flow compressibility is significant~compare cases 2and 9!. An examination of different components of the tubulent kinetic energy for cases 2 and 9 indicates that wthe dilatational component increases by the reaction thelenoidal component decreases. Our results~not shown! alsoindicate that the effects of reaction on the dilatational turlent energy and the thermodynamic variables are depenon the initial flow compressibility. In the cases in which thinitial variations in temperature and density are more signcant, the variation in the reaction rate would also be msignificant and the thermodynamic variables as well asdilatational turbulent motions are affected more by the hof reaction.
1. Energy transfer
To examine the effects of the reaction on the enetransfer between the internal (EI5reI) and the kinetic (EK
5reK) energies, the transport equations for^EI& and ^EK&are considered,
d
dt^EI&52^PD&2^VD&1^HR&, ~13!
d
dt^EK&5^PD&1^VD&, ~14!
where
^PD&[ K p]uj
]xjL ,
FIG. 12. Temporal variation of the mean kinetic energy for different cas
,
-
t-Inre
n
tl
he
leo-
-nt
-eet
y
^VD&[ K Q i j
]ui
]xjL ,
^HR&[Ce
~g21!Mo2 ^wP&,
are the ‘‘pressure-dilatation,’’ the ‘‘viscous-dissipation,and the ‘‘heat-release’’ terms, respectively.37 Examination ofEqs. ~13! and ~14! indicates that the heat of the reactiondirectly transferred to the internal energy and the meannetic energy may only be modified indirectly through tvariations in the pressure-dilatation and viscous-dissipaterms. The total energy (ET5EI1EK) is only affected bythe heat of reaction and is constant in non-heat-releacases. In the heat-releasing cases considered here, the vof the internal energy are much larger than those of thenetic energy and monotonically increase by the heat of retion.
The rate of variation of the mean kinetic and internenergies, as discussed above, are dependent on the predilatation, viscous-dissipation, and heat-release terms.tailed examination of each of these terms helps us expthe results in Figs. 10–12. In the heat-releasing cases conered in this study, the magnitudes of the heat-release termmuch larger than those of the pressure-dilatation aviscous-dissipation terms and the internal energy variesmarily due to this term. The heat of reaction does not haany direct influence on the kinetic energy. However, the rof change of the kinetic energy is controlled by the pressudilatation and the viscous-dissipation terms. Both of theterms as shown in Fig. 13 are affected by the heat of retion. In the non-heat-releasing case 1, the values of the psure dilatation oscillate around zero, indicating that theergy is alternatively transferred between the internal andkinetic energies via this term. In this case, the pressudilatation term is relatively small as the flow is nearly incompressible. However, consistent with the results of Balakrinanet al.,36 Jaberi and Madnia;37 and Martin and Candler,38
the amplitude of the oscillation of the pressure-dilatatiterm is significantly increased by the heat of reaction. Fig13~a! shows that the values of this term in cases 3 and 4an order of magnitude larger than those in case 1. Compsons of the results for cases 1 and 5 and cases 3 aindicate that the variation in molecular coefficients has liteffect on the evolution of the pressure dilatation.
Temporal variations of the viscous-dissipation termdifferent cases are shown in Fig. 13~b!. In non-heat-releasingcase 1, this term has the dominant effect and decreases~in-creases! the kinetic~internal! energy. With the decay of turbulence, the gradients of the velocity and the magnitudethe viscous-dissipation term decrease. Nevertheless, thenitudes of this term in heat-releasing case 3 are considerlarger than those in non-heat-releasing case 1. This is dumodification of the molecular coefficients and the smascale turbulence by the heat of reaction. The most significdifference between the results in cases 1 and 3 occur1.5,t,3, when the reaction is very important. A compason among the results in cases 1, 3, and 4 indicates thamagnitudes of the viscous-dissipation term increase pri
s.
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1200 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
rily due to increase in the molecular coefficients. The vometric expansion/contraction of the fluid elements aslightly increases the magnitudes of the viscous-dissipaterm ~compare cases 1 and 4!. Interestingly, the long timevalues of^VD& in case 3 are smaller than those in casedespite the fact that the magnitudes of the molecular cocients in case 3 are much larger than those in case 4explain these observations, it is useful to compare the evtion of ^VD/m& for cases 3 and 4. In case 4,^VD/m&5^VD&. Figure 13~b! shows that the magnitudes of^VD/m&in case 3 are significantly lower than those in case 4. In c3, the values ofm are higher, the ‘‘smoothing’’ effects of themolecular viscosity on the velocity gradients are more iportant, and the magnitudes of^VD/m& decay faster. Thisexplains why the long time values of^VD& in case 3 aresmaller than those in case 4.
The influence of the heat of reaction and the flow copressibility on the viscous-dissipation and the pressudilatation terms is further assessed in Fig. 14, where the tintegrated values ofPD& and ^VD& for several differentcases are considered. The variation in the mean kineticergy is equal to the summation of these integrated quantiFigure 14~a! shows that in non-heat-releasing cases the ingrated values of the pressure-dilatation term are relativsmall. In the heat-releasing cases, these integrated valuepositive and significant. While flow compressibility has litteffect in non-heat-releasing cases, it significantly amplifithe effect of reaction on the integrated values of the presdilatation in heat-releasing cases~compare cases 1, 2, 6, an9!. The positive sign of the integrated values of the pressudilatation term indicates that this term on the average
moves energy from the internal energy and adds it tokinetic energy. In the exothermic reacting flows, as thesults in Fig. 14~a! suggest, the role of the pressure-dilatatiterm is very important and should be considered in the meling of these flows, particularly when the flow compressibity is significant.
In contrast to the pressure dilatation, the integrated vues of the viscous-dissipation term are always negativeincrease in magnitude as heat release increases@Fig. 14~b!#.In non-heat-releasing cases, the magnitudes of this quadecrease as the initial flow compressibility increases, whis consistent with the results shown in Fig. 12. However,heat-releasing cases the flow compressibility has an oppoeffect and enhances the magnitude of the viscous dissipaThis is primarily due to an increase in the dilatational turblent motions~see the discussion corresponding to Fig.below!.
It is shown above that the turbulent kinetic energy athe terms responsible for its evolution are noticeably affecby the heat of reaction. However, the results in Figs. 10 a11 indicate that the solenoidal and the dilatational comnents of the turbulent energy are affected differently byreaction. It is, therefore, useful to consider the evolutiequations for the solenoidal and dilatational energy comnents. Analysis of these transport equations also helpbetter understand the energy transfer process in reacflows. The interactions between different modes of thenetic energy and the internal energy in compressible floare studied in detail by Kida and Orszag,44 and Jaberi andMadnia.37 They decomposeWi5Arui into the mean, the
FIG. 14. Time-integrated values of different terms in the mean kineticergy equation@Eq. ~14!#, ~a! the pressure dilatation,~b! the viscous dissipa-tion.
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1201Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
rotational, and the compressive components. Similar analis conducted here. The decomposition of the kinetic eneand also the governing equations describing the evolutionthe compressive and rotational components of the kineenergy for constant molecular transport coefficients are pvided by Kida and Orszag44 and are not given in detail hereIt is only adequate to present the evolution equations ofspatially averaged values of the kinetic energy componeThese equations are written as
d
dt^~EK!b&5^~AD!b&1^~PD!b&1^~VD!b&, ~15!
where (EK)b , b[R,C,O denotes the rotational, the compressive, and the mean components of the kinetic enerespectively. In Eq.~15!, ^(AD)b&, ^(PD)b&, and^(VD)b&represent the effect of the advection, the pressure dilatatand the viscous dissipation on the volumetric averaged vues of the kinetic energy components and are defined as44
^~AD!b&[ K S 2uj
]Wi
]xj2
1
2WiD DWb i L ,
^~PD!b&[K 21
Ar
]p
]xiWb i L ,
~16!
FIG. 15. Temporal variation of the rotational and the compressive comnents of~a! the advection term,~b! the pressure-dilatation term, and~c! theviscous-dissipation term.
isy
ofico-
ets.
y,
n,l-
^~VD!b&[K S 2
ReoAr
]
]xjFmS Si j 2
1
3Dd i j D G D Wb i L ,
whereWb i ,b[R,C,O denotes the rotational, the compresive, and the mean components ofWi , respectively. Ourresults~not shown! indicate that in both non-heat-releasinand heat-releasing cases the mean components of the ation, ^(AD)O&, the pressure dilatation(PD)O&, and the vis-cous dissipation,(VD)O&, are negligible as compared to thcompressive and rotational counterparts.
Temporal variations of the rotational and the comprsive components of the advection term for both non-hereleasing and heat-releasing cases are considered in15~a!. This figure shows that in the absence of heat releathe compressive and the rotational components of the adtion term fluctuate symmetrically with respect to the horizotal ~zero-value! axis and there is no net contribution to thkinetic energy by these components. In the case with conerable heat release, again the compressive and the rotatcomponents fluctuate symmetrically around the horizonaxis but the amplitude of their oscillations is larger than thin the non-heat-releasing case. Additionally, during the tiperiod that the reaction is significant, the time averaged vues of^(AD)R& and ^(AD)C& are positive and negative, respectively. These results are consistent with those of Jaand Madnia37 and indicate that on the average, the energytransferred from the compressive component of the kinenergy to its rotational component. To explain this behavit is useful to consider the mechanisms responsible forchange in the dilatational fluid motions. In compressible noreacting flows, the compressibility effects caused by initconditions or other factors, such as shock waves, enhancdilatational turbulent motions.44 The solenoidal fluid motionsmay also amplify the dilatational fluid motions through thadvection term. Alternatively, the energy could be tranferred from the dilatational motions to solenoidal motionsthe dilatational advection which is the case in our nonreaing simulations. In reacting flows, the heat of reaction mofies the dilatational fluid motions. The modified dilatationfield then may affect the solenoidal field. The direction of tenergy transfer between the solenoidal and the dilatatiocomponents of the kinetic energy depends on the rate ofrelease and the energy residing in each component. In cathe energy released by the reaction is noticeably largethe net energy transfer is from the dilatational componenthe solenoidal component.
The temporal variations of the rotational and the copressive components of the pressure dilatation for caseand 3 are considered in Fig. 15~b!. It is the pressure-dilatation term that alternatively transfers energy from tinternal energy to dilatational and solenoidal parts ofkinetic energy and vice versa. The results in Fig. 15~b! showthat in the non-heat-releasing case, the rotational compoof the pressure dilatation is negligible, indicating that tpressure dilatation does not transfer energy to or fromrotational kinetic energy. In the case with considerable hrelease, the values of^(PD)C& and^(PD)R& oscillate aroundzero but the amplitude of oscillations is significantly high
o-
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1202 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
than in the nonreacting case. Again, consistent with thesults of Jaberi and Madnia,37 the compressive component othe pressure dilatation has magnitudes substantially lathan those of the rotational component. The cases whigher initial flow compressibility exhibit behavior similar tthat shown in Fig. 15~b!. The variation of the pressuredilatation term in Figs. 13~a! and 14~a! with reaction andflow compressibility is mainly due to variation of its dilatational component as the solenoidal component is not notably affected. The dilatational component of the pressurelatation changes in magnitude with reaction and flcompressibility.
The influence of reaction on different components ofviscous-dissipation term is shown in Fig. 15~c!. In the ab-sence of heat release, the magnitude of the rotationalcompressive components of the viscous dissipation decrwith turbulence decay. In this case, the magnitudes^(VD)R& are much larger than those of^(VD)C&, suggestingthat the dissipation scales are controlled by the vortical mtions. Nevertheless, Fig. 15~c! shows that both the rotationaand the compressive components are significantly affeby the reaction, although the compressive component isfected more. It is observed that the magnitudes of^(VD)R& in heat-releasing case 3 are higher than thosenon-heat-releasing case 1 att,4. The increase in solenoidadissipation is primarily due to the increase in molecularefficients with temperature, since the results for cases 1 aare close. The increase in dilatational dissipation is dueboth enhancement of the small-scale dilatational velofluctuations and increase in molecular coefficients. A coparison between the results in cases 1, 3, and 4 indicatesthe magnitudes of(VD)C& increase by almost an order omagnitude with reaction, even if the molecular coefficieare kept constant. Additionally, the magnitudes of^(VD)C&decay slowly as compared to those of^(VD)R&. Conse-quently, the long time values of the rotational and the copressive dissipation become comparable. Our results~notshown! also indicate that the temporal evolution of^(VD)R&is not very much dependent on the initial flow compressibity in non-heat-releasing cases. However, the values^(VD)C& are higher and are enhanced more by the heareaction in cases with higher initial flow compressibility.
D. Enstrophy
The results shown in Fig. 11 indicate that different scaof the solenoidal and the dilatational velocity fluctuationsaffected differently by the chemical reaction. While alength scales of the dilatational velocity field are amplifiethe small-scale solenoidal motions are primarily affectedthe reaction. An important quantity which characterizes thsmall-scale solenoidal motions is the enstrophy. The temral variation of the mean enstrophy for several different cais shown in Fig. 16. In the absence of heat release,^V& de-cays monotonically~and almost exponentially! due to vis-cous dissipation. However, the heat of reaction influencesmean enstrophy andV& decays much faster in heat-releasicase 3. To isolate the effect of reaction on the molecuviscosity coefficient from other factors that affect^V&, in Fig.
e-
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e
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16~a! the temporal evolution ofV& for cases 4 and 5 is alsconsidered. The results corresponding to these cases clindicate that the values ofV& are affected by the heat release, primarily due to variations in the molecular transpcoefficients. A comparison between cases 1 and 3 in F16~a! and 10~b! indicates that the effect of reaction on thenstrophy is more significant than that on the solenoidalnetic energy. This is understandable since the small~dissipa-tive! flow scales are affected the most.
In contrast to dilatational turbulent motions, the smascale solenoidal motions are affected similarly by the retion for different initial flow compressibility. This is demonstrated in Fig. 16~b!, where the decay of the mean enstropfor cases 1, 2, 6, and 9 are considered. Figure 16~b! is con-sistent with Fig. 16~a! and shows that the reaction increasthe decay rate of the mean enstrophy, regardless of the inflow compressibility. As mentioned before, initially the summation of the dilatational and solenoidal turbulent energiethe same in all cases. But in cases 2 and 9, the initial vaof the solenoidal energy and enstrophy are lower than thin cases 1 and 6. Nevertheless, the long time values ofmean enstrophy in cases 1 and 2 and also those in casand 9 are very close. The solenoidal kinetic energy, althoaffected less by the reaction, exhibits a qualitatively simibehavior. These results suggest that in reacting and nonreing flows the long time values of the ‘‘solenoidal statisticsare independent of the initial flow compressibility. Of cour
FIG. 16. Temporal variation of the mean enstrophy for different case
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1203Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
they are affected by the reaction, primarily due to variatioin molecular coefficients.
1. Enstrophy transport equation
To understand how the chemical reaction affects the vticity field and to explain the results in Fig. 16, the termsthe enstrophy transport equation are examined. The transequation for mean enstrophy reads as
~17!
wherev andS= are the vorticity vector and strain rate tensorespectively. The solenoidal and the dilatational viscoforces are defined as
j s[¹•~mS= s!,~18!
j d[¹•S mS= d2m
3DI= D ,
whereS= s and S= d denote the solenoidal and the dilatationstrain rates, andI= is the identity tensor, respectively. TermsII, III, IV, and V on the right-hand side~rhs! of Eq. ~17! areidentified as the vortex-stretching, the vorticity-expansithe baroclinic, the solenoidal-dissipation, and tdilatational-dissipation terms, respectively. The vortestretching term is responsible for energy transfer amongferent turbulent scales and usually has a positive sign.vorticity-expansion term can have a positive or negative ctribution to the mean enstrophy, depending on the correlabetween the contraction/expansion regions of the flowthe local values of the enstrophy. The baroclinic techanges the mean enstrophy only if the pressure gradienthe density gradient vectors are not aligned. The solenoand the dilatational viscous terms decrease the magnitudmean enstrophy. In low Mach number nonreacting flows,dilatational-dissipation term is usually negligible.
Temporal evolution of all terms on the rhs of Eq.~17!for cases 1 and 3 are shown in Fig. 17. In the non-hereleasing case 1@Fig. 17~a!#, the magnitudes of the vorticityexpansion, the baroclinic, and the dilatational-dissipatterms are relatively small and the mean enstrophy machanges by the vortex-stretching and the solenoiddissipation terms, as expected for a nearly incompressflow. The vortex-stretching term has a positive sign andmagnitudes are slightly lower than those of the solenodissipation. As a result, the mean enstrophy decays conously. The vorticity-expansion and the baroclinic termscillate around zero. In the heat-releasing case 3@Fig. 17~b!#,the vortex-stretching and the solenoidal-dissipation termsstill the dominant terms and decrease in magnitude withbulence decay. However, a comparison between the re
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nlyl-leslu--
rer-lts
in Figs. 17~a! and 17~b! indicates that these terms decafaster in heat-releasing case 3. This is due to higher valuethe molecular coefficients in the heat-releasing case. Alsothe heat-releasing case, the magnitudes of terms II, III, anare comparatively higher than those in the non-heat-releacase. The baroclinic term is positive and peaks whenmean reaction rate reaches its maximum value. The vorticexpansion term has magnitudes comparable to the barocterm but contributes both positively and negatively to tmean enstrophy. The magnitude of the dilatationdissipation term also peaks when the mean reactionpeaks but is lower than that of the vorticity-expansion abaroclinic terms.
The integrated values of all terms on the rhs of Eq.~17!are shown in Fig. 18 for various cases. The summationthese integrated quantities is equal to the change in the mnitude of the mean enstrophy. The results in Fig. 18consistent with those in Fig. 17, indicating that all termsthe rhs of Eq.~17! are affected by the heat release. A comparison between the results for cases 1 and 3 in Fig. 1~a!indicates that the integrated value of the vortex-stretchterm is significantly decreased by the reaction and reachnearly constant value att.4. This is primarily due to varia-tion of the molecular coefficients with temperature, asresults for cases 1 and 4 are very close. In contrast tovortex-stretching term, the integrated values of the vorticiexpansion and baroclinic terms increase substantially byheat of reaction. The values of the vorticity-expansion teare lower when the molecular coefficients are kept cons~compare cases 3 and 4!. The baroclinic term exhibits opposite behavior as its integrated values are increased morenificantly by the reaction if the molecular coefficients a
FIG. 17. Temporal variation of different terms in the mean enstrophy traport equation@Eq. ~17!# for ~a! case 1,~b! case 3.
in
1204 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
FIG. 18. Time-integrated values of different termsEq. ~17! contributing to the mean enstrophy,~a! term I,~b! term II, ~c! term III, ~d! term IV, ~e! term V.
dr
arhe
r-
ea
ts
ane
thaut
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rm
-ictheebyon
n
ig-calherm,thant.entse in-
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kept constant. This indicates that the increase in magnituof the molecular coefficients weakens the baroclinic genetion of the mean enstrophy.
The solenoidal- and the dilatational-dissipation termsalso affected differently by the heat of reaction. While tmagnitudes of the solenoidal dissipation decrease by theaction @Fig. 18~d!#, those of the dilatational dissipation increase@Fig. 18~e!#. Figures 18~d! and 18~e! also show thatthe magnitudes of the~solenoidal-! dilatational-dissipationterm in case 4 are~higher! lower than those in case 3. So, thvariation of the molecular coefficients with temperature hthe opposite effect on these two viscous terms. The resulFig. 18~d! are also consistent with those in Fig. 18~a! andindicate that the magnitudes of the solenoidal-dissipationthe vortex-stretching terms decrease by reaction due tohancement of the molecular coefficients. In contrast,magnitudes of the vorticity-expansion and the dilatationdissipation terms increase as the magnitudes of the moleccoefficients increases. It is to be noted that even thoughvorticity-expansion, the baroclinic, and the dilatationdissipation terms are strongly affected by reaction, their ctributions to the mean enstrophy remain much less than thof the vortex-stretching and the solenoidal-dissipation te
esa-
e
e-
sin
dn-el-larhe
-ses
in all cases. Our results~not shown! suggest that in exothermic reacting flows while the magnitudes of the baroclinand the dilatational-dissipation terms are dependent oninitial flow compressibility, the vortex-stretching and thsolenoidal-dissipation terms are not noticeably affectedthe compressibility. Therefore, the effects of the reactionthe mean enstrophy as shown in Fig. 16~b! are similar incases with different initial flow compressibility.
Generation of vorticity via baroclinic torque plays aimportant role in the flame-vortex interactions.48,60,61 Al-though in the flows studied the baroclinic term does not snificantly change the mean enstrophy, it has important loeffects on the vorticity field and the flame structure. In tmean enstrophy transport equation, the baroclinic te2^1/r2@v•(¹p3¹r)#& is composed of three vectors; bothe magnitude and relative alignment of these are importThe magnitudes of the pressure and the density gradiincrease as the pressure and the density fluctuations arcreased by reaction. This is shown in Fig. 19~a!, where thevolumetric averaged values of the magnitudes of the denand the pressure gradients for cases 1, 3, and 6 are coered. The mean value of 1/r2 follows closely the trends observed in Fig. 19~a! for density gradient but its magnitude
he
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1205Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
are increased by less than 15% with reaction. For the isotmal reacting cases, bothu¹ru& and ^1/r2& decrease slowlyand continuously due to turbulence decay. In exothermicacting cases, they peak at the time corresponding to preaction rate and decay later. The decrease in the magnof the vorticity vector is somewhat balanced by the increin ^1/r2& in heat-releasing cases. This decrease inuvu, asexplained earlier, is due to an increase in molecular coecients with temperature.
The second factor which influences the magnitude ofbaroclinic term is the alignment of the pressure and the dsity gradient vectors. The time evolution of the mean vaof sinj ~j is the angle between the pressure and the dengradient vectors! is presented in Fig. 19~b!. The results inthis figure clearly indicate that the reaction has a significinfluence on the alignment of these two vectors as^sinj&increases with reaction. At the time which the mean reacrate peaks, the density and the pressure gradient vectorsto be mostly perpendicular. To further examine this behior, in Fig. 20~a! the PDFs of sinj for cases 1 and 3 arcompared. The results for case 1 are consistent with thobtained by Kida and Orszag62 and indicate that the pressuand the density gradients are almost aligned. This alignmis also supported by the high correlation between the psure and the density which is a consequence of a neisentropic process. The PDF of sinj is changed by the hearelease and attains a peak at sinj'1 and very low values asinj'0. This indicates that in most of the domain the presure and the density gradients are perpendicular. Our re~not shown! also indicate that the pressure and the den
FIG. 19. Time variation of~a! the magnitudes of the density and the presure gradients,~b! average of the sine of the angle between the pressgradient and the density gradient.
r-
e-akdee
-
en-eity
t
nnd-
se
nts-rly
-ltsy
fluctuations are poorly correlated when the heat releassignificant.
Another quantity which influences the behavior of tbaroclinic term is the angle between the vorticity vector athe baroclinic torque (c52¹p3¹r). The PDFs of the co-sine of this angle (cosl) at t52.5 are shown in Fig. 20~b!.The highest probability is at61, indicating thatc andv aremostly parallel. In case 1, the PDF is nearly symmetric athe mean value of cosl is very small. In case 3, the PDF islightly skewed toward positive values at 1.5,t,3.5. Con-sequently, the mean value of cosl is positive and small.
E. Dilatation
It is demonstrated above that the dilatational turbulmotions are substantially modified by the heat of reactiAn important quantity which characterizes the small-scdilatational turbulent motion is the second moment of dilation (^D2&). This term is almost equal to the dilatation varance since the mean value of the dilatation is nearly zeFigure 21 shows the temporal variation of^D2& for variouscases. In the non-heat-releasing case, the values of^D2& arehigher for higher initial flow compressibility but decreaseall cases with turbulence decay. The results in Fig. 21also consistent with those shown in Figs. 10~c! and 11~b! andindicate that the magnitudes of^D2& increase substantially
re
FIG. 20. PDFs of~a! the sine of angle between the pressure gradient anddensity gradient,~b! the cosine of angle between the vorticity vector andc.
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ioIV
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1206 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
with the heat of reaction. This increase is dependent onrate of heat release and is more significant in caseshigher initial flow compressibility.
To understand how the dilatational fluid motions arefected by the reaction, the transport equation for^D2& isconsidered,
~19!
Temporal evolution of the terms on the rhs of Eq.~19! forcases 1, 2, and 9 are shown in Fig. 22. In case 1@Fig. 22~a!#,the flow is nearly incompressible and the magnitude of^D2&and all terms contributing to its evolution are very small adecline with the decay of turbulence. Term II representscorrelation between the dilatation and the magnitude ofstrain and tends to decrease^D2&. Term III represents thecorrelation between the enstrophy and the dilatation andappears in the enstrophy transport equation multiplied21/2. This term has both negative and positive contributbut its time integrated values are mostly negative. Termrepresents the correlation between the pressure gradienthe dilatation gradient. This term has the most significinfluence on^D2& and always enhances the fluctuationsthe dilatation. The two viscous terms have opposite effeon ^D2&. The first term~term V! represents the correlatiobetween the gradient of the dilatation and the strain rate.second term~term VI! is due to dilatation gradient and hamagnitudes slightly lower than the first term. While the vues of ^D2& decrease by the first viscous term, the secoterm increases them.
FIG. 21. Temporal variation of the second moment or variance of the dtation for different cases.
eth
-
ee
soyn
ndtfts
e
-d
A comparison between cases 1 and 2 in Figs. 22~a! and22~b! reveals the effects of compressibility on terms on trhs of Eq.~19!. While in case 1 the values of term III arcomparable to other terms, they are relatively insignificancase 2. This suggests that the correlation between the ditional and the solenoidal velocity fields is small and is nsignificantly affected by the flow compressibility. The flocompressibility has, however, a significant effect on othterms. For example, the magnitudes of term I in case 2much higher than those in case 1. In both cases this terma negative contribution. The behavior of term II is also vedifferent in cases 1 and 2. This term decreases the fluctions of the dilatation in case 1. In the cases with significinitial compressibility~case 2!, the reverse is true and termhas the most significant positive contribution. The magnituof term IV increases substantially by the flow compressibity, as expected. The viscous terms V and VI exhibit simibehavior in cases 1 and 2. Term V always contributes netively to the values ofD2& but the contribution of term VI isalways positive. The magnitudes of term V are larger ththose of VI resulting in a net negative contribution to tdilatation variance by the viscous terms.
It is shown in Fig. 21 that the values of^D2& are signifi-
-
FIG. 22. Temporal variation of different terms in Eq.~19! contributing tothe second moment of dilatation~a! case 1,~b! case 2,~c! case 9.
1207Phys. Fluids, Vol. 12, No. 5, May 2000 Characteristics of chemically reacting compressible . . .
cantly affected by the heat of reaction. It is, therefore,surprising that all terms contributing toD2& are also af-fected by the reaction. In fact, Fig. 22~c! shows that with theexception of term III the magnitudes of all terms on the rof Eq. ~19! increase by more than an order of magnitude wreaction. Nevertheless, the behavior is similar in non-hereleasing and heat-releasing cases 2 and 9, indicatingterms II and V have the most significant positive and netive contributions, respectively. Close examination of thesults in Figs. 22~b! and 22~c! indicates that the values of termIII in cases 2 and 9 are small and comparable. This agsuggests that the vortical and the dilatational velocity fieare weakly correlated.
IV. SUMMARY AND CONCLUSIONS
Direct numerical simulations are conducted of chemcally reacting homogeneous compressible fluid flow unnon-heat-releasing~isothermal! and heat-releasing~exother-mic! non-premixed reacting conditions. The chemistrymodeled with a one-step irreversible reaction and with acoefficient of an Arrhenius type.
Examination of the compositional flame structure incates that the finite rate chemistry~or temperature dependency! effects are important and the reaction rate exhibbehavior different than the flame surface density. Duringtime that the reaction is significant, the ‘‘mixing’’ term (G5r2YAYB) and the ‘‘temperature-dependent’’ term@F5exp(2Ze/T)# have comparable mean values. Also, at ttime the mixing term is not well correlated with the reactirate. While the values of the temperature-dependent termhigh at the ‘‘reaction zones,’’ those of the mixing term adependent on the spatial density variations and are oftenin these zones. Additionally, in the reaction zones the flowdominated by strain rather than rotation and the scalardient is mostly aligned with the most compressive eigenvtor of the strain rate tensor. Consistent with the previoobservations, the heat of reaction decreases the alignmbetween intermediate strain eigenvector and vorticity vecparticularly near the reaction zones.
The results of simulations with isothermal reaction areaccord with the previous findings and indicate that the flundergoes a nearly isentropic process and the pressuredensity fluctuations are very well correlated. However,exothermic reacting simulations, the dilatational~compres-sive! turbulent motions at all length scales are significanintensified by the heat of reaction. The effect of reactionthe dilatational velocity field is enhanced as the initial flocompressibility~the initial fluctuations in dilatational velocity and thermodynamic variables! increases. Analysis of thetransport equation for dilatation variance (^D2&) indicatesthat the magnitudes ofD2& and almost all terms contributing to its evolution increase substantially with increase ininitial flow compressibility and/or the heat of reaction. Thexothermicity of the reaction also increases the fluctuatiof the thermodynamic variables at all length scales.
In contrast to the dilatational velocity field, the low ordmoments of the solenoidal~rotational! velocity field and thescalars are not significantly affected by the reaction. Thi
t
s
t-at--
ins
-r
te
se
s
re
wsa--sentr,
nd
n
e
s
is
because the large-scale solenoidal velocity field is notrectly affected by the reaction and the solenoidal and dilational velocity fields are poorly correlated. However, tsmall-scale rotational turbulent motions and the related qutities such as the enstrophy are noticeably influenced byheat of reaction. This is primarily due to variation of thmolecular transport coefficients with temperature andvolumetric flow expansion/contraction has lesser effeAnalysis of the enstrophy transport equation indicates tthe effects of baroclinic torque increase as the heat releand/or the initial flow compressibility increases. Neverthless, the contributions of the baroclinic torque and the vticity expansion are much less than those of the vortstretching and the solenoidal viscous-dissipation.
Examination of the energy transfer among differemodes of the kinetic energy and the internal energy in ethermic reactive flows indicates that the energy of the retion is transferred to the compressive component of thenetic energy by the compressive component of the pressdilatation correlations. The advection term then transfersenergy from the compressive component of the kineticergy to its rotational component. The compressive androtational components of the turbulent advection andcompressive component of the pressure-dilatation exhsignificant oscillations in time. The amplitude of these osclations enhances due to the heat of reaction. The comprescomponent of the viscous dissipation also increases in mnitude as a result of the energy transfer from the interenergy to the compressive component of the kinetic eneThe effects of reaction on pressure dilatation and viscdissipation increase with the flow compressibility duestrong coupling between the ‘‘turbulence-generated’’ and‘‘heat-generated’’ dilatational fluid motions. Also, in acases considered, the rotational and the compressive comnents of the kinetic energy are poorly correlated.
The results presented in this paper reveal the intricphysics of the two-way interactions between turbulencechemical reaction. The future models of the turbulent reaing flows have to account for these interactions. We arelizing the DNS results obtained from this work to develonew subgrid scale models in Large Eddy Simulations~LES!of turbulent reacting flows. We are specifically interestedhow the subgrid stresses and scalar fluxes and the cosponding models in LES are affected by the chemical retion. The behavior of the subgrid unmixedness in reactflows is also of interest and is being studied using the datthis work.
In this study, the turbulent Mach number and the flocompressibility are relatively small. The turbulence is adecaying due to lack of any mean velocity gradient or sheAnalysis of the flame-turbulence interactions in highly copressible and supersonic homogeneous and inhomogenchemically reactive turbulent flows would be the next chlenging tasks.
ACKNOWLEDGMENTS
This work was sponsored by the National Science Fodation under Grant No. CTS-9623178. Computational
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1208 Phys. Fluids, Vol. 12, No. 5, May 2000 Jaberi, Livescu, and Madnia
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