characteristics of a subgrid model for turbulent premixed

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Characteristics of a Subgrid Model for Turbulent Premixed Flames*

V. K. Chakravarthy * £ S. MenontSchool of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, GA 30332-0150

AbstractA modified version of the linear eddy model (LEM) forsubgrid combustion is developed and used for studyingthe properties of turbulent premixed flames in the coreregion of Couette flow. The handling of the burning(diffusion/reaction) and specie transport, in this new ap-proach, is fully deterministic. The model is, therefore,found to predict the correct laminar flame speed in thelimit of zero turbulence. The effect of three dimensionaleddies on scalar fields at the subgrid scales is modeledusing a stochastic Lagrangian rearrangement procedure.The derivation of this procedure from the inertial rangescaling laws necessitates a calibration procedure sincethe numerical values of the constants in these laws areunknown. The present LEM approach relates the con-stant needed to the coefficient of eddy viscosity that isevaluated in a LES, under the assumption of constantturbulent Prandtl and Schmidt numbers. The new ap-proach is found to predict the turbulent flame speed withfair amount of accuracy. The chemistry is modeled us-ing a self propagating scalar field (G-equation) approachand within the limitations of this model, LEM is found tocapture the right trends in evolution of flame structure,effects of heat release and turbulence. The encouragingresults obtained here warrant further research into ex-tending this subgrid model for cases of finite rate chem-istry and inclusion of thermodiffusive mechanisms suchas the Lewis number effects.

1 IntroductionThe simulation of turbulent phenomena in engineeringis, in the present day greatly limited by the available

t Professor, Senior Member, AIAA.* Graduate Research Assistant.* Copyright ©1997 by V. K. Chakravarthy & S. Menon. Pub-

lished by the American Institute of Aeronautics and Astronautics,Inc., with permission.

computational resources. This neccesitates the develop-ment of modeling tools that would enable us to simulatethese phenomena within the realm of available resources.One such approach is Large Eddy Simulation (LES) inwhich the large scale structure of the flow is determinedby solving the governing equations that are augmentedwith additional (approximate) model terms to accountfor the effect of small scales. Dissipation of kinetic en-ergy is the only physical phenomenon of importance innon-reacting flows that occurs at small scales. Separat-ing these small scales and the large scales (of the sizeof the characteristic geometric length in the flow) is theinertial range which acts purely as an energy cascadefor supplying kinetic energy to the small scales from thelarge scales. The near universal behavior of the iner-tial range along with the assumption of local small scaleisotropy has aided the modeling of turbulent flows usingthe eddy viscosity assumption.

However, there is no universally valid form of thescalar spectrum at small scales. There exist several formsof the of scalar spectrum depending on the local Schmidtnumber (for species) or Prandtl number (for temper-ature). Independent of these parameters, scalar mix-ing using an eddy diffusivity, nevertheless, is often usedin the present day modeling studies. This approach isseverely in error in cases where scalar counter gradientdiffusion occurs. Further, if the fluid is reacting, theinteraction between the molecular mixing and chemicalreactions occuring at smallest of the scales (Kolmogorovand Batchelor scales) have significant effects on largescale fluid/flame structure. The additional non-linearityin the governing equations in case of reactive systemsalso needs to be handled perhaps with greater care foraccurate modeling of reacting flows.

In many practical engineering systems, combustiontakes place in a regime of parameter space called the cor-rugated flamelet zone. The flame thickness is very smallas compared to any relevant fluid dynamic length scales


(Kolmogorov length scale). In this region of parameterspace, the flame is wrinkled by turbulence but the lo-cal burning is laminar. The turbulent flame consists ofan ensemble of locally laminar flames whose self prop-agation speed (laminar flame speed) is governed by thebalance between temperature and specie diffusion. Theprobability density function (PDF) and moment closureapproaches are, at the present day, inadequate in han-dling the diffusion processes and hence it is difficult topredict the correct reaction rate in the subgrid whenthese approaches are used.

When the reaction system is dominated by one physi-cal phenomenon, there are several models that work well.Yakhot's model [1] and Fractal models [2](for thin pre-mixed flames with low or no heat release) that concen-trate on the kinematic structure of the flame and charac-terize the model in terms of incoming turbulence, Furebyand Moller model [3] which uses the turbulent reactionrate based on local turbulent and chemical time scales,are a few examples. Approaches such as these could beused to arrive (often empirically) at simple inexpensivemodels that, on calibration over certain regions of pa-rameter space, could work for specific class of flows.

The Linear Eddy Mixing (LEM) model seeks to sep-arate and model independently the effects of molecularmixing from the turbulent effects on the flame. Theimplementation is on a one-dimensional domain whichconsiderably reduces the modeling cost. By carryingthese one-dimensional domains in each of the LES cellsto account for the effects of unresolved turbulence onscalar transport and production/destruction, this ap-proach has been extended for use in LES [4]. The lin-ear eddy model(LEM) is further capable of accountingfor the thermodynamic effects such as volumetric dilata-tion, differential diffusion, viscosity variation with tem-perature etc. There is yet another important featureof this model that is very advantageous in modeling re-acting flows once the advection (due to resolved scales)is modeled correctly. That is the adequate numericalresolution of the flame. Thin flames produce steep gra-dients in flow properties that could become very difficultto handle computationally if one uses finite volume or fi-nite difference schemes. If not adequately resolved, mostnumerical schemes cause unphysical oscillations near theflame that threaten to destabilize the numerical solution.Generally used robust schemes usually are dissipative innature and smear out the physical gradients, thus, artifi-cially thickening the flame. The LEM approach handles

the scalar transport in a Lagrangian manner circumvent-ing this problem.

The Linear eddy model has been used in several nu-merical combustion studies. The kinematic structure ofpremixed flames was modeled earlier by Menon et al.[4]and this approach was extended for modeling the kine-matics of a self propagating scalar interface in a wallbounded shear flow using LES with dynamic A'-equationmodel [5]. This study utilized the G-equation [6] whichmodels the motion of self propagating kinematic surface(model for cold flame). A subgrid kinetic energy equa-tion model with dynamic evaluation of the model co-efficients is used as the subgrid model for momentumtransport. This study proves the feasibility of an onlineimplementation of the LEM approach in a 3D LES butasks for further research into species advection modelingand inclusion of heat release effects. Some inadequaciesin the advection algorithm were overcome by Calhoonand Menon [7] in their two dimensional simulation ofreacting mixing layers including effects of heat release.However, as noted in [8], the inadequacies in this advec-tion seem to cause significant errors in premixed flamesas against non-premixed flames. A modified version ofthe advection algorithm is reported here.

The present research seeks to validate (independently)and combine the various different modeling procedures(for each of the underlying physical phenomena) to arriveat a comprehensive modeling approach for use in LES.Using this approach, the structure of premixed flamesin turbulent Couette flow at Re (based on channel widthand wall velocity difference) of 20000 is analysed. Exper-imental data from [9] is available at that Re for valida-tion of cold flow LES. The Couette flow has a wide coreregion where the mean flow is small and the turbulentintensity is nearly constant. The turbulent intensity inthis region can be used as a parameter against which theflame characteristics (like turbulent flame speed, flamestretch etc.) could be characterized. Also reported hereare the effects of heat release on the flame structuralproperties.

The flame generated turbulence is found to be a signif-icant factor in determining the flame speed. The precisenature of it's effects needs to be studied can be studiedusing a stationary flame problem. To this end, a pre-mixed turbulent stagnation point flame is under studyusing LES. Some preliminary observations are reportedhere.

An eddy viscosity approach based on the model A'-


equation (subgrid kinetic energy equation), discussed insection 2, is used to model momentum transport. A zeroMach number fractional step method is used to numer-ically integrate the LES equations on a non-staggeredgrid. The version of the LEM approach used here is out-lined in section 3. The results are analysed in section 4and directions for future research are discussed in section5.

has in it terms accounting for production, transport anddissipation. It is a representation of the temporal non-equilibrium of the subgrid scales because the productiondoes not neccesarily equal the dissipation (even on anaverage).

A variable density version of the model (for use inzero Mach number equations) is used here. The eddyviscosity and subgrid dissipation are given as follows.

2 Governing equations for LESThe Navier Stokes equations on convolution with a spa-tial filter (after filtering), reduce to the following LESequations.

dpdt+P = 0

dt

(1)

(2)

The volumetric dilatation dUi/dxi is obtained fromthe zero Mach number approximation of the energyequation arrived at by assuming a fast acoustic time scaleapproximation from the equations. The physical conse-quence of this is to remove any gradients in thermody-namic pressure (because of infinite propagation speed).The normal stress associated with the momentum gra-dients (as in incompressible flows) is however spatiallydependent.

For a closed set of equations , one needs to approx-imate the subgrid stresses using a model. The veloc-ity variations in the scales below the characteristic filterwidth A are unresolved in a LES. Due to the nonlin-ear nature of the Navier-Stokes equations, these smallscale fluctuations effect the large scale motions. This ef-fect comes from the subgrid stress, which in the presentstudy is approximated as r,-;- = —^pK6,j +2^j5,j, whereSij = | [dUi/dxj + dUj/dxi] is the resolved strain ten-sor, lit is subgrid eddy viscosity (to be defined later)and A' = — \(Ui Ui — U,Ui) is the subgrid kinetic en-ergy. Filtered variables are also called supergrid vari-ables because they carry information about a variablesat all length scales above the filter width (grid spacing).

A A'-equation (for the subgrid kinetic energy) model[10] is used as the subgrid model. The advantage of thismodel is that it solves a single scalar equation for thesubgrid kinetic energy which characterizes the velocityscale of subgrid turbulence. The subgrid kinetic energy

esgs = Cf

(3)

(4)

For the transport term, a gradient diffusion modelbased on eddy diffusivity model (with unit eddy Prandtlnumber) has been proposed and studied by Menon andKim [11]. This approximation was found to adequatelymodel the transport terms. Hence, this is used in a sim-ilar form in this study. The dynamic equation for A" cannow be written as:

dpKdt

(dpKUj) _———-—————— — TV'

The turbulent dissipation due to compressibility anddilatational effects is assumed to be insignificant at thelow Mach that are to be encountered. Cv and C( are themodel constants that need to be specified. These con-stants, however, are not universal and differ with flow-fields in general. This suggests that these constants alsodepend on the local (supergrid) structure of the flowfield. It is, then appropriate to refer to them as co-efficients rather than constants. A dynamic approach isapplied here to evaluate these coefficients, thus removingthe arbitrariness in prescribing these coefficients. Theapproach is based on the concept of subgrid stress simi-larity supported by experiments in jets (Liu et al. [12]).In this approach, a test filter (similar to the LES filter) ofcharacteristic width 2A is defined and the correspondingfiltered velocity field is denoted by Ui. This new veloc-ity field is obtained by convolution of the LES filteredvelocity with the test filter. The subgrid stress corre-sponding to the scales in between the grid filter widthand the test filter width is termed the test filter stress.Assuming stress similarity between the test filter stressand the subgrid stress, along with equations similar toeq.(3) and eq.(4) the test filter variables, one can arrive


at equations for the model coefficients. These equationscan be found in [13] and are omitted here for brevity.

The equations are discretized on a non-staggered grid(with spacing corresponding to the characteristic filterwidth A) and numerically integrated using a two stepsemi-implicit fractional step method. The Poisson equa-tion in this method is solved numerically using an ellipticsolver that uses a four-level multigrid scheme to con-verge the solution. The species transport between cellsis handled in a Lagrangian manner and thus the speciesequation need no integration on the supergrid level.

3 Linear Eddy subgrid modelA subgrid modeling approach based on linear eddy mix-ing model (Kerstein, [14-17]) was proposed by Menon etal. ([18],[19],[20]). In this approach, the turbulent stir-ring of the scalar field and molecular diffusion (leading tolaminar propagation) are accounted for distinctly. Theeffects of each of these phenomena on flame characteris-tics can hence be studied in isolation. The modeling pro-cedure with appropriate justification is presented here inbrief.

The subgrid modeling is conducted on a one-dimensional domain (in each of the LES cells) represent-ing a ray of length A across the turbulent flame brush.This domain is divided into number (specification to beexplained later) of cells each representing a finite vol-ume with the total volume adding upto the volume ofthe LES cell.

A field equation for self propagating scalar interfacecalled the G-equation [6] is used to model the laminarflame propagation in this study. This equation is de-signed to propagate a scalar interface at the laminarflame speed in stagnant flow and has the following form.

For the case of moving fluid, the interface also getsconvected along with self propagation and a convectionterm needs to be added to the above equation.

The scalar G here, is a representation of the rate of ad-vancement in a premixed reaction. It has a binary repre-sentation on the subgrid one-dimensional domain (laterrefered as the linear eddy domain). G = 1 representsan unburnt state and G = 0 represents a burnt state oneach LEM cell. The interface between two dissimilar ad-jacent cells is considered to be an infinitely thin flame

that propagates into the unburnt zone at a specifiedflame speed. The temperature rise due to the reactionis modeled by assuming the temperature to be a linearfunction of G. In this way, the G-equation replaces thereaction-diffusion equations for both species and tem-perature (that, in conjunction make the premixed flameself propagating). In the burnt state, the temperaturerises to the prescribed value of the product tempera-ture. Since the thermodynamic pressure is assumed tobe constant (which is the case across premixed flamesgoverned by second order chemical kinetics), the densityvaries inversely with temperature. Since the transportand diffusive properties of G and temperature would beclosely correlated in this approach, it is to be assumedthat these simulations correspond to a case of constantLe (Lewis number), specifically 1.0.

As can be seen in the above equation, the G-equationdoes not include any convective term. The effect of sub-grid velocity (turbulence) is modeled separately using astochastic process. A Lagrangian rearrangement is usedto model the effect of fluid dynamic eddies on the scalarfield. The length scale of the eddy (segment size on whichthis rearrangement process is to be conducted) and thefrequency of stirring at that length scale are determinedusing isotropic inertial range (assumed to exist in thesubgrid scales of the LES) scaling laws. There is nouniversality of the rearrangement scheme. It can be ar-bitrary but it should mimic the effect of hydrodynamiceddies on the scalar field and be conservative (retain thelowest scalar moments invariant).

A triplet map [17] is chosen for the present study. Inthis mapping, the chosen segment is divided into threeequal parts. All spatial gradients are increased by a fac-tor of 3 in the portions at the ends, and a factor of-3 in the middle portion. The scalar field in the threesegments is pieced together to give a continuous scalarfield with scalar values at the end points of the segmentremaining the same as before and the scalar gradientsincreased by 3 times in magnitude (as illustrated in [5]).These increased gradients cause increased scalar diffu-sion in accordance with what has been observed as theeffect of turbulence on scalar transport and dispersion.The number of finite volume cells in each subgrid domainis so chosen that a stirring event at the smallest relevant(energetic possessing) scale could be implemented. Thisis assumed to be the Kolmogorov length scale t]. A linesegment needs to have atleast 6 cells to implement atriplet map, so the cell size should be no less than 77/6.


So the total number of cells is 6A/VThe triplet map causes particle dispersion with mean

square displacement of 4/2/27, where / is the length ofthe segment. The diffusivity associated with such a pro-cess would be 2A/3/27, where A is the frequency per unitlength. The diffusivity due to turbulence for all lengthscales below / can be determined from the inertial rangescaling laws (relevant dimensional analysis) in terms ofthe rj, I and v. It can be shown that it has the same formas eddy viscosity at that length scale, only difference be-ing the constant of proportionality is now Cv/Prt in-stead of Cu • Cv is determined as explained in the earliersection and it is assumed that Prt is 1.0. Thus the con-stant of proportionality is arrived at from analytical rea-soning rather than a calibrating procedure undertakenin [7]. Equating the value of diffusivity to the corre-sponding integrated effect of triplet mappings at variouslength scales, one can arrive at the PDF for length scale /and the stirring frequency (using the procedure adoptedin [5]). The stirring procedure in conjunction with themolecular diffusion has been shown to effectively modelthe effect of three dimensional inertial range turbulenceon the scalar fields [21].

Advection due to the supergrid velocity brings abouttransport of chemical species from one LES cell into it'sneighbouring cells. This is modeled using a proceduretermed as "splicing" [17]. The scalar flux across each LEScell face is computed using the filtered velocity on the cellface. The number of subgrid cells corresponding to thescalar flux are transfered from one cell to it's neighbouracross the cell face in accordance with the direction ofthe scalar flux.

Instead of picking these cells from a random locationon the donor subgrid domain like in the earlier research[5], cells are tranfered from one end of the subgrid do-main. The cells that are spliced in are put at the oppositeend of the domain. In time, the linear eddy cells to bespliced in first get spliced out first. In the laminar limit(acheived by removing stirring), the flame travels fromone end to another end of the cell at the rate of Si, thelaminar flame speed. When a LES cell is fully burnt theflame crosses over to the next LES cell and this processis found to predict the right laminar flame speed on thesupergrid (on an average). The earlier splicing algoritmwith randomly chosen locations for transfer of subgridsegments does not ensure this. Further, the introduc-tion of cells spliced into the receiver subgrid domain cancause extra flames at the interface (between the cells put

in and the existing cells). The splicing algorithm mod-els advection and is not supposed to create any scalargradients (flames in the present case) on the linear eddydomain. So, in the event of an artificial flame being cre-ated at the interface, the spliced in cells are rearranged toremove this extra flame. The only exception to this pro-cedure is the case of burning initiation in a fully unburntcell in which case one flame caused at the interface is re-tained. This procedure prevents the existence of morethan one flame in laminar flows (there is no flame brushin laminar flames). This approach was found necessaryto obtain the correct laminar propagation on the super-grid especially in case of curved flames (cylindrical andspherical). In the turbulent case, the stirring could causemultiple flames to exist on the same subgrid domain andthis could be interpreted as a stochastic representationof flame brush in each cell.

Finally the average density along the subgrid domainis used as the filtered density on the supergrid and is usedto compute the volumetric dilatation on the supergrid.

4 Results and Discussions.The LES numerical code is validated against the experi-mental results obtained from turbulent Couette flow ex-periments[9]. Shown in fig. 1 and fig. 2, respectively, arethe profiles of mean velocity and rms of axial velocity forCouette flow at Re of 20000. A 49x49x33 is utilized forthis simulation. Hyperbolic tangent stretching is usedfor the grid spacing along the wall normal direction. Ascan be seen, the mean velocity profile is anti-symmetricabout the centre-line where as the experimental predic-tion is not. It was argued that this does not cause anybias in the u profile which is in fact found to be sym-metric. The peak in u profile is much closer to the wallsin experimental data than in the LES. The cause of thisdiscrepancy is unknown. It is likely that the wall layerneeds more resolution or the experimental results maybe in error due to moving wall (belt) vibrations whichmake the flow noisy. This LES approach has earlier beenvalidated by simulating turbulent Couette flow at Re of5200 by comparing with experiments in [5].

In order to ease the implementation of LEM on LESgrid, the mid 78 percent of the grid is made uniform.The simulations are stopped once the flame reaches theboundaries of this uniform region. The capability of lin-ear eddy subgrid model in capturing thin flames is il-lustrated in fig. 3. For comparison, starting with the


same initial conditions, LES is conducted using a finitedifference implementation of G-equation with turbulentflame speed obtained using the Yakhot's model [1]. Ninecontours corresponding to nine equally spaced values ofG are shown in fig. 3. The finite difference implementa-tion of chemistry is seen to thicken the flame front, thus,reducing the gradients. This in turn causes a reductionin flame speed. As was sought from the conclusions ofearlier LEM research [5], the linear eddy model (with thenew advection modeling approach) now propagates thelaminar flame on the supergrid at the correct flame speedfor both planar and spherical flames. This removes thearbitrariness in advection algorithm [5] and the calibra-tion procedure for diffusion/burning [7]. The turbulentstirring becomes the only stochastic part of the approachand the only arbitrariness is in prescribing Prt.

The premixed flame propagation is simulated for val-ues of u /Si : 1.0, 4.0 and 8.0. For all these cases, simu-lations are conducted for three different values of prod-uct temperatures (Tp). Tp/Tf of 1.0, 4.0 and 7.0 areused. So there are nine cases in the parameter space.Since the flames are not stationary, one needs several re-alizations of the flow at the same time after ignition forstatistical analysis. For each of the above cases, severalLES realizations (starting from different realizations ofnon-reacting flow LES as initial conditions before ignit-ing the flame) of the flow are simulated till the statisticsconverge.

The temporal evolution of the flame structure is dis-cussed first, followed by analysis of effects of increasingrelative (to 5;) turbulent intensity. Since the cold flameshave been studied earlier, this analysis is done for flamesin the presence of heat release. The effects of increasingheat release are discussed later.

The local structure of the flame can be characterizedusing the principle radii of curvature. These are com-puted from the eigen values of the curvature tensor [22].The ratio of smaller radius to the higher, is refered toas the shape factor. It lies between -1 (correspondingto a saddle point) and 1 (spherical shape). The valueof 0 corresponds to a cylindrical shape. The hydrody-namic effects on the flame are mainly characterized bytwo quantities, the tangential strain rate in the planeof the flame and the flame stretch which is defined asthe rate of change of a Lagrangian flame surface at agiven location. Mathematical means of computing thesequantities can be found in [22]. The strain rate gives anestimate of the hydrodynamic force acting in the plane

of the flame. The flame stretch determines the rate atwhich the flame is deforming with the effects'of self prop-agation included. Positive stretch indicates a tendencyof the flame to replanarize thus reducing the curvatureand hence also the turbulent flame speed. In the limitof a material surface (5; = 0), the tangential strain rateand the flame stretch are highly correlated. Probabil-ity density functions (PDFs) of each of these local flameproperties are constructed in order to characterize thegross nature of the flame. The strain rate and flamestretch are non-dimensionalized using the Kolmogorovtime scale and the curvature is non-dimensionalized us-ing the integral length scale (prescribed).

In fig. 4, the transition from a spherical to (locally)cylindrical nature of the flame is shown for the case ofu /Si of 8.0 and Tp/T} of 7.0. When the fuel is ignited ata point source, the initial growth is (globally) spherical.With increasing time the effect of turbulence is realizedin terms of wrinkles that are characterized by increase inshape factor PDF near cylindrical zone. Eventually theeffect of mean velocity comes in and elongates the flameand the turbulence wrinkles the flame front. This is il-lustrated graphically to a certian extent in fig. 5. Thenthe flame becomes mostly cylindrical and finally, towardsthe end of the simulation, some saddle nature is noticed.This trend of PDF shifting towards the saddle zone frommostly cylindrical nature (once the effect of turbulenceis fully realized) is also noticed as the shear layer evolvesspatially in [22]. In fig. 6 is shown the evolution offlame stretch PDF with time. The time here is non-dimensionalized by the total time of the simulation (justbefore the flame leaves the domain where LEM is im-plemented). For an outward burning hot flame, the cur-vature and volumetric dilatation (expansion) are alwayspositive leading to positive flame stretch. As the flameexpands, the curvature effect reduces and turbulence ef-fect is felt more. In isotropic turbulence, the stretch isnearly symmetric except that the self propagation contri-bution skews the PDF slightly towards the positive side.This skewness reduces with increasing u /Si and PDFis very nearly symmetric for material surfaces [23]. Theflame is indeed found to transition towards this naturegradually from it's initial state of fully positive stretch.

The simulations with Tp/Tj of 7.0 are analyzed tostudy the effects of reducing Si with same u (increasingu /Si). The tangential strain rate PDFs for three casesof u /Si are shown in fig. 7. Increasing the flame speedcauses the negative curvature to decrease and positive

6


curvature (concave to the fuel side) to increase. Due topositive volumetric dilatation on the burnt side, regionsof positive curvature experience high strain rates on thesurface. So the PDF is more skewed to the right for thecase of higher 5;. Alternately, as turbulence intensityincreases relative to Si, the nature of the flame tendsmore towards that of a material surface thus reducingthe skewness. As noticed in earlier studies [22] involvingspatial mixing layers, the mean curvature PDF (fig. 8)is slightly shifted to the right (because of finite Si) andis found to be independent of u /Si. The flame stretch(fig. 9) is found to behave in much the same fashionas the tangential strain rate because the curvature re-mains unchanged. Because of slight positive skewness incurvature PDF, it is easy to see from the expression forflame stretch, that the flame stretch skewness is slightlyreduced as compared to flame strain rate.

The relative alignment of the flame normal with thedirection of most compressive strain rate (eigen vectorcorresponding to the lowest eigen value of the strain ten-sor) is computed and plotted in fig. 10. Contrary to theprevious reports [24] of the PDF peak at an alignmentangle of 30° in shear driven flows, the peak is closer to 0°indicating a tendency of the flame normal to align withthe compressive strain rate direction. This was observedin isotropic turbulence simulations earlier [24]. Two rea-sons could perhaps be attributed for this behavior. First,the mean shear is very small in the core region and hencethe turbulence may be close to being locally isotropic.Second, in the presence of heat release, the mean sheareffect may be insignificant because the flow is modifieddrastically due to density changes. The volumetric di-latation tries to accelarate the flame propagation thustrying to compress the flame brush. In such cases, thehigher two eigenvalues are both most likely positive thusmaking the tangential strain rate highly positive for highheat release cases as noticed in fig. 7.

The effects of increasing heat release on the flamestructure are analyzed using cases with u /Si of 4.0. Thetangential strain rate is plotted, for the three values ofheat release under consideration, in fig. 11. As seen ear-lier, the most compressive principle strain rate alignsclosely with the flame normal at most of the points.The sum of the three principle strains is positive andincreases with increasing heat release. Since the strainrate normal to the flame is negative, it is very likely thatthe two higher values of the principle strain rates arepositive leading to an overall positive strain rate in the

plane of the flame. Mean curvature (shown in fig. 12)is slightly biased towards the positive side/ The likeli-hood of regions on the flame with higher mean curvatureis found to increase with heat release. The heat releasecauses faster flame growth due to dilatation but this ac-celeration could dewrinkle the flame, thereby reducingthe local radii of curvature (increasing curvature). Theflame stretch is plotted in fig. 13 and is found to corre-late well with the behavior of the tangential strain ratesince the curvature is nearly symmetric.

The turbulent propagation rate of the premixedflames, while they are in the core region, is character-ized against the turbulent intensity of the cold flow intowhich the flame is propagating. The turbulent flamespeed predictions (experimental and theoretical) exhibita large amount of scatter. The key reason for this beingthe fact that there may be parameters that are differentin various studies and, hence, comparing the results fromthese to arrive at some definite conclusions cannot berationalized. Bradley et al. [25] in their survey of avail-able of experimental results for characterizing turbulentflame speed against inflow turbulence, found it necessaryto consider the dimensionless Karlovitz stretch factor asa relevant parameter. It is a measure of chemical timeagainst the eddy time and is given as [26]:

K = ;T)

where R\ is the turbulent Reynolds number based on theintegral length scale.

The results reported were therefore separated into dif-ferent zones depending on the value of this parametertimes the Lewis number. They report results that fallinto two zones corresponding to mean values of K.Levalues of 0.1 and 1.0 with a scatter of about 25 precent.Some of these experimental results are plotted along withthe predictions from the current study in fig. 14. Thevalues of the above mentioned parameter are 0.01, 0.15and 0.46 respectively for u /Si values of 1.0, 4.0 and 7.0.Also included are the best fit lines to experimental pre-dictions at K.Le of 0.1 and 1.0. It is found that thecold flame front propagation matches the experimentaldata rather well. However, the turbulent flame propa-gation is much faster in cases with heat release. Thecompounded effect of two factors could be the reasonfor the observed increase with heat release. Flames thatare primarily convex (burning outwards from burnt side)experience some amount of outward accelaration. The


countering effect would be that this increases the flamestretch thus reducing the flame speed. Also, a wrinkledexothermic propagating front generates some fluid dy-namic diturbances which in turn change the flame prop-agation speed. These disturbances (termed flame gener-ated turbulence) add onto the incoming turbulence andone can define an "effective turbulence intensity" [26]that determines the flame structure. Leisenheimer andLeuckel [27] were able to measure this effective intensityin a premixed flame propagating radially. Radial sym-metry (in their study) on an average allowed for decom-position of the flow into an average field and turbulence.There is no way of studying this phenomenon using anarbitrary non-stationary flames and is left as an objec-tive for future research.

The LEM approaches is being currently used to sim-ulate turbulent premixed stagnation point flows. Thisflame is stationary and also is two-dimensional on anaverage. This presents one with opportunity to study,statistically, several phenomena in turbulent combustionat a lowered compututional expense. A 65x49x33 gridis being used to simulate this flow on a domain that is70mm x 200mm in wall normal and lateral directions.The width in the third direction is set at 70mm. Theinflow velocity is 5.0m/s with 10.0 percent turbulence.These parameters are being used with an aim of simu-lating the flames studied in [27].

With the conditions being used, the turbulent flamespeed turns out to be about 4 times the laminar flamespeed (for Si = u') which is unusually high. It is foundthat this could be reduced by decreasing the frequencyof stirring (when put to zero, the LEM predicts the lam-inar flame speed). The reason for stirring being higherthan required is that the subgrid kinetic energy that de-termines the frequency, is probably not being predictedcorrectly. The inflow subgrid is set at 0.1 times the tur-bulence level rather arbitrarily and this may be in error.

The increased stirring rate in LES of stagnation pointflame is found to reduce the wrinkling of the flame on thesupergrid. Some 3-dimensional wrinkling is neverthelessseen in fig. 15. Contours corresponding to G level of 0.5are plotted for several locations along the homogeneousdirection in fig. 16. Instantaneous burning is found tooccur in supergrid cells that are within two grid pointsdistance from the mean position of the flame. This in-dicates that the LEM approach is able to capture theflame brush in just a few grid prints.

5 ConclusionsIn summary, it can be concluded that the modified ad-vection algorithm overcomes the inadequacies noted inearlier research. The linear eddy subgrid approach, inthe absence of turbulent stirring, is found to model lam-inar flame propagation correctly. In the presence of stir-ring (the only non-deterministic part of the approach),it is found to also predict the turbulent flame speedwith fair amount of accuracy. The trends in evolution ofthe spatial structure of the premixed flames are also inagreement with experimental and DNS predictions. Thepresent approach is however limited by the G equationand the corresponding state equation for temperature.The LEM needs to be extended to flames involving realchemical kinetics in order to account for effects of ther-modiffusive mechanisms. Also intended for future re-search is the subject of flame generated turbulence. Thepreliminary results from LES of premixed flames in stag-nation point flames are found to be quite encouraging. Infact, this problem could provide the platform necessaryfor studying flame generated turbulence.

AcknowledgementsThis research was supported by the NASA Lewis Re-search Center under Grant no. NAG3-1610.

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Figure. 1 : Mean velocity profile in turbulent Couette flow.Solid line : LES, symbols : experiments [9]

0.2

0.2

1 0.1

0.1

0.00.0 0.5

V/h1.0

Figure.2 : Axial rms velocity profile in turbulent Couette flow.Solid line : LES, symbols : experiments [9]

(i)

(ii)

Figure.3 : G-level surface contours.(I) Finite difference method, (ii) LEM

8.0

6.0 -

LLQ 4.0o.

2.0

0.0

- time = 0.2--- time = 0.4---- time = 0.6

O——Gtime = 0.8A——A time =1.0

-1.0 -0.5 0.0 0.5Shape factor

Figure.4 : LEM predicted temporal evolution of flamestructure.


Figure.5 : Geometry of the flame surface.(i)t = 0.6, (ii)t = 1.0

time = 0.2time = 0.4time = 0.6time = 0.8time = 1.0

30.0

20.0

U.QDL

-0.05 0.00 0.150.05 0.10Flame stretch

Figure.6 : Evolution of flame stretch in time.

0.20

Q0.

250

200

150

100

50

0-0.03 0.00

Mean curvature

*=*0.03

Figure.8 : Effect of laminar flame speed (relative to aconstant turbulence level) on the mean curvature.

10.0

0.00 0.05 0.10 0.15Tangential strain rate

Figure.7 : Effect of laminar flame speed (relative to aconstant turbulence level) on the tangential strain rate.

0.20


20.0 4.0

3.0

LLQ 2.0Q_

1.0

0.00.00 0.10 0.20

Flame stretch0.30 0.40 0.0 0.2 0.4 0.6 0.8 1.0

Figure.9 : Effect of laminar flame speed (relative to aconstant turbulence level) on the flame stretch.

40.0

30.0

Q 20.0Q.

10.0

Figure. 10 : Effect of laminar flame speed (relative to aconstant turbulence level) on the flame normal alignment

relative to the most compressive strain direction.

400

-0.10 0.00 0.10Tangential strain rate

0.20

Figure. 11 : Effect of heat release on the tangential strainon the flame surface.

-0.02 -0.01 0.00 0.01 0.02mean curvature

Figure. 12 : Effect of heat release on mean curvature oftheflame surface.

25.0

20.0

15.0

Q0-

10.0

0.0 0.1 0.2 0.3Flame stretch

Figure. 13 : Effect of heat release on flame stretch.

25.0

20.0

15.0

c/)

10.0

5.0

0.00.0


2.5 5.0 7.5 10.0 12.5u'/S,

Figure. 14 : Comparison of turbulent flame speed predictions with experimental data.(K = Karlovitz stretch factor [26], Le = Lewis number)

—— K.Le = 0.1— — K.Le = 1.0

o Abdel-Gayed et al.(1985)A Abdel-Gayed et al.(1988)n Andrews et al.(1975)+ Smith etal.(1978)• T/T, = 1• T/T, = 4 [gfent

AT/T.-7

Figure. 15 : 3-dimensional structure of the flame in stagnation point Figure. 16 : Superposition of G=0.5 (flame) contoursflow. along the homogeneous direction and the

instantaneous streak lines.

characteristics of a subgrid model for turbulent premixed

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