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

AIAA Meeting Papers on Disc, January 1997A9715437, AIAA Paper 97-0372

Large-eddy simulations of turbulent reacting stagnation point flows

Thomas M. SmithGeorgia Inst. of Technology, Atlanta

S. MenonGeorgia Inst. of Technology, Atlanta

AIAA, Aerospace Sciences Meeting & Exhibit, 35th, Reno, NV, Jan. 6-9, 1997

A methodology for solving unsteady premixed turbulent flame propagation problems in high Reynolds number (Re), highDamkoehler number spatially evolving flows is developed. The method is based on Large-Eddy Simulation (LES) with asubgrid combustion model based on the Linear-Eddy Model (Kerstein, 1991). An inter-LES cell burning mechanism hasbeen added to the present formulation to account for molecular diffusion (burning) across LES cells which was beenneglected in earlier studies. Two-dimensional simulations of passive isotropic flame propagation using the LES-LEM for Refrom 100 to 1000 are conducted to validate the subgrid and supergrid propagation mechanisms. Two models are used tosimulate turbulent premixed stagnation point flames. The first model is a conventional approach which uses a thin flamemodel (Kerstein et al., 1988) with and without significant heat release. The second model uses the LES-LEM approachwithout heat release. Qualitative comparisons between simulations and experimental data are made. (Author)

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Large-Eddy Simulations of Turbulent Reacting Stagnation PointFlows *

Thomas M. Smith^and S. Menon*School of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, GA 30332-0150

AbstractA methodology for solving unsteady premixed tur-bulent flame propagation problems in high Reynoldsnumber (Re), high Damkohler number (Da) spatiallyevolving flows is developed. The method is based onLarge-Eddy Simulation (LES) with a subgrid combus-tion model based on the Linear-Eddy Model (Kerstein,1991). An inter-LES cell burning mechanism has beenadded to the present formulation to account for molecu-lar diffusion (burning) across LES cells which was beenneglected in earlier studies. Two-dimensional simula-tions of passive isotropic flame propagation using theLES-LEM for Re from 100 to 1000 are conducted tovalidate the subgrid and supergrid propagation mech-anisms. Two models are used to simulate turbulentpremixed stagnation point flames. The first model is aconventional approach which uses a thin flame model(Kerstein ti al, 1988) with and without significant heatrelease. The second model uses the LES-LEM approachwithout heat release. Qualitative comparisons betweensimulations and experimental data are made.

1 IntroductionNumerical procedures for calculating turbulent react-ing flows at high Reynolds numbers (Re) encounteredin engineering situations have been mainly limited tosteady state Reynolds Averaged Navier-Stokes (RANS)approximations. In RANS, the effects of all turbulentscales on the mean quantities are modeled simultane-ously, typically using gradient transport assumptions

•Copyright ©1997 by T. M. Smith and S. Menon. Publishedby the American Institute of Aeronautics and Astronautics, Inc.,with permission.

tGraduate Research Assistant, Student Member, AIAA'Associate Professor, Senior Member, AIAA

to compute terms arising from Reynolds-averaging thegoverning equations. In some cases, probability densityfunctions (pdf's) are used to describe the interactionsbetween chemical reactions and turbulence. Whilequantities such as the mean fuel consumption rate andthe mean heat release may be estimated by RANS,transient flame-turbulence interactions such as local ex-tinction can not be captured in a steady state calcula-tion. Therefore, the need still exists for a methodologythat is capable of capturing transient and unsteady phe-nomena in turbulent reacting flows.

The difficulty in simulating unsteady combustionproblems lies in the problem of resolving the enormousrange of active scales encountered in high Re and highDamkohler numbers (Da is defined as the ratio of acharacteristic flow time to a chemical time scale) re-acting flows. In high Da flows chemical reactions oc-cur in thin sheets. This increases the requirement forresolution far exceeding the requirement of resolvingthe turbulent scales alone. Since the flame interactswith turbulence at the smallest scales, the small scaleinteractions can not be neglected. Direct NumericalSimulation (DNS) is an ideal method from the stand-point that no closure models are required since all activescales are resolved on the computational grid. DNS ofpremixed freely propagating flames in isotropic turbu-lence (Trouve and Poinsot, 1994) and through isolatedvortex pairs (Poinsot et al, 1991) have provided an in-creased understanding of flame-turbulence interactionsincluding Lewis number effects and a mechanism for lo-cal extinction. Although it remains a valuable researchtool, present and foreseeable computational capabilitiesrestrict DNS to moderate Re and low Da flows.

Large Eddy Simulation (LES) is a promising tech-nique that bridges the gap between the steady stateassumption made by RANS and the intractable com-prehensive DNS approach, In LES of turbulent non-

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reacting flows, the energy-containing large scales aredirectly simulated and the subgrid scales (assumed tobe locally isotropic) are modeled.

Models based on the subgrid kinetic energy can beused (Kim and Menon, 1996) to compute the subgridstress in the filtered equations. Turbulent flows may besimulated with adequate accuracy and at a fraction ofthe computational expense compared to DNS.

LES of turbulent reacting flows has received less at-tention mainly due to the lack of adequate subgrid com-bustion models. However, recent studies have proposedthe adoption of the Linear Eddy Model (LEM) of Ker-stein (1991,1992) (Menon et al., 1994; Calhoon andMenon, 1996; 1997) as a subgrid combustion model.LEM is a stochastic mixing model that simulates scalardiffusion and turbulent mixing (convection) separatelyand accounts for all relevant scales of motion explic-itly. For premixed flames, this means that the one-dimensional flame structure is resolved and all scales offlame wrinkling are explicitly captured by the model.The LES-LEM uncoupled approach has been demon-strated on transient premixed flame propagation inmixing layers (Menon et al., 1994) and a coupled pro-cedure has been developed for non-premixed reactingmixing layers (Calhoon and Menon, 1996; 1997).

This paper focuses on LES of premixed turbulentcombustion with an emphasis on subgrid combustionmodeling. To this end, a coupled LES-LEM proce-dure is developed for premixed turbulent combustionin the laminar flamelet regime and its specific appli-cation to stagnation point flows is discussed. Severalissues unique to premixed combustion LES of turbu-lent stagnation point flames using a thin flame modelwith and without heat release are also presented.

The stagnation point flow has been chosen for severalreasons. First, the flow field is stationary so the flameachieves a stationary position. A premixed turbulentjet impinging on a flat plate produces a diverging flowand a decelerating mean velocity. Therefore, a flamepropagating upstream into the premixture and awayfrom the plate will encounter an increasing velocity.As a result, a flame perturbed upstream will be con-vected back toward the plate while a flame perturbeddownstream toward the plate will burn faster than thelocal fluid velocity and will propagate back upstreamuntil a balance between flame speed and fluid velocityis reached. The stationary nature of the flame providesa convenient means for gathering statistical data neces-sary for evaluating the subgrid combustion model andfor analyzing the properties of the flame. Along withstationary propagation, the flame propagates normal to

the impinging jet, a situation that simplifies modeling.In addition, an unambiguous turbulent flame speed canbe defined. This configuration has been studied exten-sively experimentally (Cho et ai, 1986; Cho et al., 1988;Liu and Lenzc, 1988; Shepherd et al, 1990; Shepherdand Ashurst, 1992; Yahagi et al., 1992) and. therefore, adatabase for comparison is avalable. Finally, the meanstrain rate of the flow, and therefore, the flame surface(in the plane of the flame) strain rate can be controlledby systematically varying the mean jet inflow veloc-ity. The effect of flame stretch (due to strain rate) onflame speed can then be systematically studied. Sub-grid model response to a varying strain field will providean important test of the model's capabilities.

The following section describes the LES numericalprocedure and thin flame model followed by a descrip-tion of LEM including recent developments and resultsfrom stand alone LEM calculations of freely propagat-ing flames. These results demonstrate the potential ofthe LEM as a subgrid combustion model for premixedcombustion in different regimes. The procedure forLEM as a subgrid combustion model is then discussed.Results from passive simulations of flame propagationin homogeneous turbulence at high Re are presented.The simulation of stagnation point flows using a con-ventional approach and the LES-LEM approach is thendiscussed followed by results from simulations and, fi-nally, conclusions are made.

2 Model FormulationA model for the computation of unsteady premixedcombustion processes is presented in this section. Themodel is made up of the filtered equations of motionand supplementary sub-models for the propagation ofreacting fronts.

2.1 LES EquationsIn LES, the flow variables are decomposed into the re-solved scale and the subgrid scale components. Thelarge scales are computed explicitly while the effects ofthe subgrid scales on the large scales are modeled. Thelarge scales contain most of the energy and the smallscales primarily dissipate energy transferred from thelarge scales. Dynamics of large scale motion are dic-tated by the geometry of the flow field and the Reynoldsnumber, while the small scale motion is relatively unaf-fected by the geometry except near walls. The subgridlength scales and times scales are small compared to

the large scales and, thus, the small scale motion ad-justs faster and are considered more isotropic than thelarge scales. These concepts have been used to developsubgrid models.

Following Erlebacher et al. (1987), the flow variablesare decomposed into the resolved scale and unresolvedscale (subgrid scale) components by a spatial filteringoperation such that / = / + /", where the (~) denotesresolved scale and the double prime (") denotes subgridscale quantities. The Favre filtered variable is dennedas / = pf/p where the overbar represents a spatialfiltering which is defined as

_dT

(1)

F is the filter kernel, D is the domain of the flow andA is computational cell width in each spatial direction.The filtering operation is normalized by requiring that

(2)

It follows from eqs. (1) and (2) that for equallyspaced and weakly stretched grids, df/dt = df/dt andd f / d x i = df/dxi. Filtering removes the high wavenumber range of Fourier components of the flow vari-ables and separates the resolved scale components fromthe subgrid scales. In this study, a box filter is im-plicitly assumed which is appropriate for finite volumeschemes. The filter function F takes on the values

~ \ 0 otherwise.

Contrary to the more traditional Favre temporal aver-aging, / ^ / and, in general, /' ^ 0. Applying filteringto the Navier-Stokes equations results in the followingLES equations for a single component fluid (Smith andMenon, 1996a)

(7)

The subgrid closure terms are given by

T-f3 - p(u^Uj - UiUj] (8)

\piTi-pUi}

= \UjTji -UjTji\.

(9)

(10)

In the above equations, fi(T) is the molecular viscosity,K = cPn/Pr, Pr is the Prandtl number and cf is thespecific heat at constant pressure. Combustion is mod-eled using a thin flame model as discussed below. Thiseliminates the need to solve the filtered species equa-tion as discussed later. The pressure is determined fromthe filtered equation of state, p = fiRT where constantmolecular weights and specific heats are assumed (as-sumptions that will be used in simulations discussed inthe next section) . The filtered total energy per unit vol-ume is given by pE — pe + ^puiui + ip[upui — w j« f ] , andthe filtered internal energy is given by e = cvf + hj .

In order to solve this system of equations, the sub-grid terms T??' , H^3' , and

form Dksf/s = Csp(kS3s)3/2/A.g where Afl is a character-istic grid size and vt is the subgrid eddy viscosity givenby vt = Cv(k'3°)l/2Ag. The constants appearing in theabove equations, Ck — 1.0, and Cv(xi,t), Ci(xi,t), aresolved dynamically (Kim and Menon, 1996).

The fe^-equation is solved simultaneously with therest of the flow equations. With k"9" and v-t determined,the subgrid shear stresses are evaluated as

(12)

where 5,-j = ^(diii/dxj + 8uj/8xi) is the resolved scalestress tensor. The subgrid energy flux is approximated

°3 = -CepvtdH>tgs (13)

where H is the filtered total enthalpy, H = cpT +IU/MI + k'a> + hj. The model constant, Ce - l/Prtwhere Prt is the turbulent Prandtl number, is heldconstant. The subgrid closure term cr'^' and third or-der correlations arising as a result of this modeling ap-proach have been neglected.

2.2 Thin Flame Propagation ModelIn high Da premixed combustion fast chemistry resultsin very thin flame fronts. The thermo-diffusive charac-ter of a propagating thin flame remains unchanged bythe turbulent flow and therefore can be modeled as apropagating front. To simulate premixed combustionas a propagating surface, the thin-flame model of Ker-stein ei al. (1988) is used in which a progress variableG is defined that evolves according to the equation

dG dG .dG .

(15) can be used to determine uj for a given fuel mix-ture. Equation (15) is a nonlinear equation and directapplication requires the use of an iterative procedure.Therefore, to reduce computational cost, a look-up ta-ble is first generated and then MJ is linearly interpolatedfrom the table in the simulations.

2.3 Linear-Eddy Model Applied to Pre-mixed Combustion

In this section the Linear-Eddy model is developed forflamelet propagation through homogeneous turbulence.Recent results from stand-alone computations are in-cluded to show the model's capabilities to capture keyflame-turbulence interactions. Modifications to includefinite-rate and thermo-diffusive effects have been car-ried out in previous studies (Smith and Menon, 1996b;1996c).

The LEM is used to fully characterize the effects ofturbulent diffusion on the reaction-diffusion processesin the flame zone. To resolve all the length scales,the computational domain is restricted to one dimen-sion which is considered to be a statistical ray throughthe local three-dimensional flame brush in the directionof mean propagation (Kerstein, 1986). The resolutionwithin this one-dimensional domain is chosen to resolveall the relevant length scales ranging from the model in-tegral length scale L to the smallest Kolmogorov eddy 77or the laminar flame thickness, 6j, whichever is smaller.

LEM incorporates turbulent stirring (convection)and laminar propagation (reaction-diffusion) separatelywith no mass averaged velocity. In the LEM, convectionis accomplished by instantaneous scalar field rearrange-ment events and cell volume expansion caused by heatrelease. These two mechanisms are discussed below.Physically, turbulent stirring increases the propagationrate by wrinkling (increasing) the flame surface whilelaminar burning acts to smooth (decrease) the flamesurface.

The LEM relates fluid element diffusivity to a ran-dom walk of a marker particle. The total turbulentdiffusion of a marker particle caused by eddies of sizeranging from L to 77 based on triplet mapping (Kerstein,1991) is given by

(16)

Turbulent stirring is modeled as stochastic rearrange-ment events which interrupt the deterministic flamepropagation. Each rearrangement event is interpreted

as the action of a single eddy on the scalar field. Threequantities govern each event: the segment (eddy) size,the location, and the rate of events. The size is deter-mined randomly from a pdf of eddy sizes

_ 5 /-8/3

W> = Ir?-5/3-!,-5/3 ^

in the range 77 < / < L (obtained from inertial rangescaling; Kerstein, 1991). The event location is ran-domly chosen from a uniform distribution within theone-dimensional domain. The event rate (or frequencyper unit length) is determined using an analogy betweenfluid dispersion in the one-dimensional domain andturbulent diffusivity (Kerstein, 1991), DT = u'L/G\,where C\ is a model constant. In earlier studies, allconstants appearing in the above noted scaling relationswere set to unity. However, for quantitative compari-son with high Re experimental data, calibration of theseconstants is required. For example, to compare LEMpredictions of scalar mixing with DNS in homogeneousisotropic turbulence, the event rate was rescaled by re-lating the LEM diffusivity to the large-eddy turnovertime in the DNS (McMurtry et al., 1993). A similarprocedure is carried out in the present study.

The event rate is determined as E = \XLEM, whereXLBM is the length of the one-dimensional domain andA is the event frequency per unit length having dimen-sions [L~lT~l] which is determined from (Kerstein,1991)

_ 54 vRe P/T?)^3 - 1]~ 5

(18)

The time interval between events is then given asAtstir = I/(A.XLEM) • Here, the Kolmogorov lengthscale is determined from the familiar inertial range scal-ing law 77 = NnLRe~3^ where Nn is an empirical con-stant.

Once the event size and location are determined andthe time of the event is reached, the rearrangementevent is implemented using triplet mapping (Kerstein,1991). This mapping first creates three copies of theselected segment and then increases the spatial gradi-ents of the copies by compressing them by a factor ofthree and reversing the middle copy. Finally, the origi-nal segment is replaced by the new mapped segment.Each mapping event requires at least six LEM cellsso that on a discretized domain, 77 is resolved by sixpoints. The mapping event has several attributes anal-ogous to turbulent convection. First, it is known thatthe flame sheet surface normal vector aligns with themost compressive strain rate direction which is mim-iced by the compressive nature of the triplet mapping.

Second, mapping increases the number of crossings ofa single scalar value which may be interpreted as anincrease in surface area caused by flame wrinkling. Fi-nally, turbulent scaling laws built into the model causethe rate of strain and rate of growth of flame surfacearea to be of the correct order of magnitude (Kerstein,1991). The ensemble of mapping events captures keymechanistic features of turbulent stirring despite thetemporally discrete representation of a time continuousprocess. Furthermore, omission of flame surface dis-placement in the other two directions (since the presentmodel is one-dimensional) has been shown to cause sig-nificant errors only at very low turbulence intensitieswhere a single flarne persists instead of multiple flames(Kerstein, 1986). Note that the component of vortic-ity in the direction normal to the flame surface doesnot create additional flame surface area nor does itwrinkle the flame. However, the other two componentsof vorticity will contribute to flame stretch, increasingthe flame surface area, which are accounted for in thismodel.

For flamelet regime combustion (81 « 77), using theG-field equation, a model for flamelet combustion us-ing the linear-eddy approach is formulated (designatedGLEM). GLEM models laminar burning by the propa-gation equation (Menon and Kerstein, 1992; Smith andMenon, 1996c)

fiC1jj- = SL\VG\. (19)

This equation tracks the propagation of a single valueof G between GjUei < Go < Gprod, where G/uei =1 and Gprod — 0. Go is a prespecified level surfacerepresenting the location of the flame. Therefore, flamepropagation is described by one scalar instead of N + 1(Menon and Kerstein, 1992). The flame speed SL isalso a prespecified constant which accounts for all of thephysio-chemical properties of the mixture. Since LEMresolves even step like fronts, no dissipation mechanismis necessary to prevent, false minima from occurring incontrast to eq. (14).

Since there is no mathematical description of expan-sion caused by heat release in eq. (19), expansion isimplemented in terms of a physical interpretation of itseffect on the G-field. Therefore, expansion is imple-mented by first prescribing a reference value Gexp thatdefines a transition from fuel to product. After eachtime step, each new cell value, G™+1, is compared tothe old value, G™. If the reference value is crossed dur-ing that time step, all the heat is released, the cell valueis set to Gprod and a number (Nexp — 1) of new Gprodcells are added adjacent to the old cell. Here Nexp is the

nearest integer ratio of T p / T j , where Tp is the producttemperature and Tj is the fuel temperature.

Results from a previous study (Smith and Menon,1996c) are included here in order to demonstrate thecapabilities of the model to capture key characteristicsof flame-turbulence interactions in homogeneous flows.Propagation of a premixed flame through an isotropicturbulent field is simulated by solving eqs. (17-19) onan equally spaced discretized line.

In fig. la, the normalized turbulent flame speed pre-dicted by GLEM and LEM (with a global finite-ratemodel) are compared to Yakhot's model predictions.LEM predictions of «t/5i agree well with the RNGprediction for the entire range in u1 /SL. Hollow sym-bols in fig. la represent individual realizations of thefan-stirred experiments and the dotted curve is the bestfit to the data given by the authors (Abdel-Gayed et al.1984). Although the LEM predictions show differencesin both magnitude and shape compared to the dottedline, they are still well within the spread of the experi-mental data.

In fig. Ib, results from simulations of four differentRe are presented. Finite-rate flames were simulated atfour different turbulence intensities, «'. The normal-ized turbulent flame speed is plotted for constant Reagainst SL/U'. The data shows agreement in the gen-eral trends seen by Abdel-Gayed ei al. (1979) at lowturbulence intensity. The curves seem to collapse asSL/U' increases. However, as «' increases, the LEMflame Ut/Si reaches a plateau and as u' increases fur-ther, UtjSL tends toward zero. This may be the resultof flame stretch. As SL decreases, the flame thicknessgenerally increases, thus, as SL/U' decreases, the ra-tio &I/TI increases and therefore, Ka increases. Eventhough in-the-plane strain rate and flame surface cur-vature are multi-dimensional effects, the LEM containselements of flame stretch by the nature of the one-dimensional therrno-diffusive curvature effects and thecompressive strain induced by triplet mapping.

2.4 Linear-Eddy Subgrid CombustionModel

In the two proceeding sections, the LES and LEMwhere described separately. In the present section, theLEM subgrid combustion model for LES is described.

The LES numerical procedure solves the unsteady,two-dimensional compressible filtered Navier-Stokesequations, the k"3S model, and the GLEM in the sub-grid using a finite-volume scheme based on the unsplitexplicit MacCormack predictor-corrector method. The

6

scheme is formally fourth-order accurate in space andsecond-order accurate in time (Gottlieb and Turkel,1976). In addition, the viscous derivatives are fourth-order accurate. The solution of the filtered density, mo-mentum, and total energy is marched time accuratelyon the acoustic time scale

LEM numerical procedure involves three subgrid pro-cesses, flame propagation, stirring, and expansion. Ineach LES cell, the subgrid scalar G is discretized on aline. Equation (19) is marched simultaneously with theLES equations. The subgrid turbulence intensity is ob-tained from the subgrid kinetic energy usgs — \J^k'ss .From usgs, a characteristic cell length, &.XLES and thelocal viscosity, the stirring properties (event frequencyper unit length and eddy length scale distribution) areobtained. Therefore, the evolution of the subgrid scalarG is coupled to the LES by the subgrid kinetic energyand temperature dependent kinematic viscosity. Toprevent numerical diffusion from contaminating the tur-bulent diffusion, the subgrid G is required to propagateas a steep front. This is accomplished by requiring thatthe discrete time step in eq. (19) equal an integer mul-tiple of the flame burning time, A.tsgs = AXLEM/SL-Therefore, G always takes on a step function 0 (burnt)or 1 (unburnt). The consequence of this discretization isthat the subgrid process &.tsgs is different from Ati^s-

Large scale convection between two adjacent LEScells is handled by splicing subgrid cells from a do-nating LES cell to a receiving LES cell (Menon et ai,1992; Calhoon and Menon, 1996). The splicing al-gorithm calculates the volume flux to be transferredacross the LES cell interface (in a finite volume for-mulation) based on the resolved velocity and subgridturbulence intensity (w,- + us^), removes an equivalentnumber of LEM cells from the donor LES cell and addsthem to the receiving LES cell. The rate of transferis based on the convective time scale of the resolvedvelocity, A

eral problems associated with conventional combustionmodel closures. First, the flame front in the subgrid istreated numerically as a discontinuity and is convectedin the subgrid with no numerical diffusion. Comparedto solving the filtered species equations, this representstrue laminar flamelet propagation. Secondly, the sub-grid eliminates the need to specify Uf, the turbulentflame speed, because it is now part of the solution.

3 Results and Discussion

3.1 DNS and LES of Isotropic Turbu-lence

The ksgs subgrid turbulence model was tested by com-paring to DNS of two-dimensional isotropic turbulence(Case I in Herring ei a/., 1974). Previous simulations(not shown) compared directly with data from their pa-per and almost identical results were reproduced. TheDNS was carried out on a 512 x 512 computationalgrid, IT; x 2ir m in dimension. The initial Re based onthe integral length scale was 383, the integral lengthwas 0.314 m and the kinematic viscosity was lxlO~3m2/s. The initial turbulence intensity was 0.1 m/s.The LES was initialized by filtering the DNS field ontoa 128 x 128 grid and the subgrid kinetic energy wasobtained by direct filtering. Figure 3a shows the initialtheoretical energy spectrum, along with results fromfour simulations on a log-log plot. The spectra weretaken at roughly four large-eddy turnover times basedon the initial large-eddy turnover time. The filteredDNS data, the LES with dynamic evaluation of coeffi-cients, and LES with no subgrid turbulence model agreewell with the DNS spectrum. Note that there is signif-icant backward transfer of kinetic energy (£) from thesmall scales to the large scales. This is characteristic oftwo-dimensional decaying isotropic turbulence. There-fore, it may be argued that a subgrid turbulence modelis not necessary for two-dimensional LES of isotropicturbulence. Further evidence of this can be observed infig. 3b. In this figure, the decay of kinetic energy andsubgrid kinetic energy normalized by the initial valuesat time t=0 s are plotted against the normalized time.While the filtered DNS kinetic energy, LES with dy-namic evaluation, and LES with no subgrid model fol-low the decay quite well, the constant coefficient modelLES is significantly different. The decay of k'9' forthe dynamic evaluation model and constant coefficientmodel are compared to the filtered DNS values. Notethat ksgs decays much more rapidly than E and there-

fore, any ksg3 in the initial field is rapidly diminished inthese two-dimensional simulations. This has importantimplications to the subgrid combustion modeling. Thismeans that kss" does not provide adequate subgrid tur-bulence intensity to model the interaction between sub-grid stirring and flame propagation. Therefore, in thefollowing results presented here, the subgrid turbulenceintensity is prespecified and held constant throughoutthe domain and time.

3.2 DNS and LES-LEM of IsotropicFlame Propagation

DNS of freely propagating flames in isotropic two-dimensional turbulence were performed on a lit x 27rusing 400 x 400 cell computational grid to provide acomparison with G-field equation flamelet propagation.The DNS solves the Navier-Stokes equations and eithera global finite-rate mechanism or the G-field equationflame model. Inflow and outflow boundary conditionswere non-reflecting and the transverse boundaries wereperiodic. The finite-rate parameters and the turbulenceparameters were similar to those used by Haworth andPoinsot (1992). A single flame was initialized in thecenter of the domain. The Re based on the initialintegral length scale was 0.416 m, the turbulence in-tensity was 0.1 m/s and the kinematic viscosity was5zlO~4ra2/s. The ratio U'/SL was initially 6.3, andTp/Tj = 4. Four snapshots from two simulations andtwo different times (normalized by the initial large-eddyturnover time, r = 1,2) are shown in figs. 4a-4d. Inthese figures, the G-field equation model is comparedto the finite-rate flames. At one and even two T, theflames look very similar as does the post flame vorticity.These results demonstrate that if enough resolution isused, the G-field equation model can mimic more com-plex combustion models. In addition, G-field equationsimulations also require significantly less computationaleffort compared to the finite-rate model simulations, forthese simulations about 33% less.

LES-LEM simulations of passive (/ero heat release)flames were conducted on a 2-Tr x 27r, 96 x 96 compu-tational grid. The ratio U'/SL was varied by increas-ing the turbulence intensity from 0.01 m/s to 0.1 m/swhile SL was held constant. The Re based on the inte-gral length scale varied from 100 to 1000. The integrallength scale was 0.4125 m, and the kinematic viscositywas 4.125xlO~5m2/s. The subgrid Reynolds numberRf-sgs was specified as 5% of the supergrid Re and allmodel coefficients were set to unity. All four boundarieswere periodic. Two flames were initialized one integral

length apart in the center of the domain. Figure 5a and5b are snapshots of the twin flames at two large-eddyturnover times for the case of u1 /Si, = 1 and 10. Thefiltered field representing the flame brush is resolved byonly one or two LES cells. The resolved flame area isvery similar in both simulations even though the u'/St= 10 is burning much more rapidly.

To study the propagation mechanisms in the LES-LEM approach, three propagation speeds are defined.The first is Ap/Ai, the resolved scale area ratio. Forthis geometry, AL is simply twice the width of the do-main. In flamelet combustion, the area ratio is ap-proximately the ratio of the turbulent to laminar flamespeed. Since these are transient simulations and nosteady area is achieved, we hesitate to call this ratio theturbulent flame speed, nevertheless, the relative magni-tudes give a good indication of what effects the modeland flow parameters are likely to have on the turbulentflame speed. The second propagation speed defined asthe subgrid turbulent flame speed (utsssjSL) is a globalaverage of the ratio of the number of flame crossingsto a single (laminar) flame crossing. The global av-erage is taken over all LES cells containing at leastone flame crossing. The third propagation speed isthe global scalar consumption rate. It is defined as/ r< Arc i p1 A/T ^ — '•IMAX

c(y) = (a(y) -

(22)

(23)

where Ay/ is the location where filtering begins andAt/ is the local grid spacing in the y direction. In allsimulations, filtering begins at a location outside thephysical domain of interest; in this case the turbulentjet width.

3.3.2 Boundary Conditions

At an inflow boundary where a time varying velocityfield is introduced, the velocity field and the tempera-ture are prescribed and a characteristic boundary con-dition is normally used to solve for the density. Pressureis obtained through the equation of state (Poinsot andLele, 1992). However, these boundary conditions werefound to exhibit unstable pressure oscillations and sonon-reflecting characteristic boundary conditions wereused, modified from Menon and Jou (1990). In thiscase the stagnation pressure and temperature are pre-scribed functions of the y-coordinate and time, the v-component of velocity is also prescribed and the u-component is computed from the two-dimensional eu-ler equations written in characteristic form and settingthe magnitude of the incoming waves to zero. This setof boundary conditions greatly reduce the amplitude ofthe reflected pressure waves while allowing for the spec-ification of an unsteady velocity inflow field. It shouldbe noted that the specification of an unsteady velocityfield results in a small unsteady velocity divergence andthis divergence tends to increase with increasing inflowMach number.

Non-reflecting characteristic boundary conditions areimposed on the vertical outflow boundaries similar tothose suggested by Poinsot and Lele (1992). The wallis assumed to be no-slip and adiabatic.

3.3.3 LES with the G-Field Equation Model

Preliminary simulations of the stagnation point flowfield and flame propagation have been made using theG-field equation flamelet model. For these conventionalsimulations, the flame speed is assumed constant andthe geometry with a curved plate is used to match theexperiments. Four simulations were conducted usinga grid resolution of 97 x 129. Two cold-flow simula-tions used a computational domain width of 0.3 m andthe two cases with heat release used a computationaldomain width of 0.5 m. It was necessary to increase

the computational domain for heat release cases be-cause of interaction between out-flow boundary condi-tions and small amplitude pressure oscillations arisingfrom unsteady heat release. The first two cases are forU'/SL — 1.33 and were conducted with and withoutheat release. The temperature ratios were Tp/Tj = 0 , 4and the inflow Mach number based on the mean veloc-ity (U = 5.0 m/s) was 0.1. The turbulent jet widthconsisted of 48 points, u' — QAm/s the integral lengthscale in the turbulent jet was 0.0037 m and Re = 80.In the second two cases, u'/SL = 1.0 and Tp/Tj = 0,7.Snapshots of two simulations are shown in figs. 7a and7b. The vorticity and flame contours are shown forU'/SL - 1-0 and Tp/Tj - 0, 7. The cold flow case showslarger scale wrinkling and a thinner flame front than theheat release case. It is also apparent from these figuresthat the hot flame interacts with the boundary layer.There is a tendency for the hot flame to be pushed closerto the wall. This effect is shown in fig. 8a which showsthe mean axial location of the four flames. Statisticswere collected for at least 5 flow through times after astationary state was reached. This trend was not no-ticed in experiments. The r.m.s. velocity componentsin the axial direction are shown in figs. 8b and 8c.Data from experiments under similar conditions shownearly constant «' and v' in the axial direction andu' fa v' fa 0.2 m/s in the region from 0.045 m to 0.07 mfor the non-reacting case and an increases in u' in theflame region of the reacting case and an increase in v'in the reacting case very near the wall. Our data forthis region show that u' slightly increases as well as v'.This is more pronounced in the reacting cases.

The mean axial velocity (not shown) is linear for thecold flow cases and show modest humps for the twoheat release cases where an acceleration due to gas ex-pansion occurs. The turbulent flame speed is estimatedby the value of the mean axial velocity where the meanscalar is 0.95. In the experiments (Cho et a!., 1986) fora range in ut/Si from 1 to 2, ut/Si. ranges between 2to 3. Our UI/SL ranges between 1.275 to 2.45 for U'/SL= 1,1.33.

3.3.4 LES-LEM Model

Preliminary results from LES-LEM simulations are pre-sented here. The stagnation wall is flat for these cases.Two simulations are conducted to demonstrate the fea-sibility of performing this type of calculation. Bothsimulations assume zero heat release, an assumptionthat will be relaxed in the near future and reportedsubsequently. The grid was 97 x 97, the distance from

10

nozzle exit to wall was again 0.075 m, the width of thecomputational domain was 0.3 m and U=5 m/s. Thetwo simulations differ by u'/SL = 1,2, Re = 80,160,and Resgs = 19,38. The subgrid time scale parameterwas set equal to the value used by Calhoon and Menon,(1996). A snapshot of the vorticity and filtered G con-tours are shown in fig. 9a for U'/SL = 2. The flamebrush is highly convoluted by the relatively large tur-bulent structures present in the flow. Since the physicaldomain of interest is roughly the jet width, the LEMsubgrid procedure is conducted only in a subdomainnear the stagnation point. In these zero heat releasesimulations, the subgrid evolution is passive and it isconvenient and efficient to solve for the evolution of theflame brush in this manner. Nearly stationary propa-gation rate data for the two simulations are presentedin fig. 9b. Though it is obvious that more data isnecessary to estimate mean quantities, it is also appar-ent that the burning rates are steady. Just as in theisotropic sirnualations, the propagation rates scale withthe u'/SL.

Though the results are only preliminary, they demon-strate that the LES-LEM procedure has the potentialfor turbulent flame propagation simulations.

4 Conclusions

A methodology for solving unsteady premixed turbu-lent flame propagation problems in high Reynolds num-ber ( R e ] , high Damkohler number (Da) spatially evolv-ing flows has been developed for combustion in the lam-inar flamelet regime. The model can easily be modi-fied to include finite-rate and thermo-diffusive effects(Smith and Menon, 1996b; 1996c). The method isbased on LES with a subgrid combustion model basedon the LEM (Kerstein, 1991). An inter-LES cell burn-ing mechanism has been added to the present formula-tion to account for molecular diffusion (burning) acrossLES cells, a mechanism previously neglected. Passiveisotropic flame propagation simulations using the LES-LEM for Re from 100 to 1000 were conducted to val-idate the subgrid and supergrid propagation mecha-nisms. Results show that the subgrid and supergridburning rates scale with U'/SL and Re. In addition,the filtered scalar that propagates as a steep front isresolved in these high Re flows with as little as one ortwo LES grid cells and subgrid flames are resolved bya single LEM cell width, reducing the numerical diffu-sion in the subgrid combustion processes. Two mod-els were used to simulate turbulent premixed stagna-

tion point flames. The first model is a conventionalapproach which uses a thin flame model (Kerstein etal., 1988) with and without significant heat release(Tp/Tf = 0,4,7). The second model uses the LES-LEM approach without heat release. Qualitative com-parisons between simulations and experimental datawere made. Turbulent cold flow statistics and turbulentflame speeds agree reasonably well with experiments byCho et al, (1986), however reacting flow statistics devi-ate in magnitude. It is believed that the differences arecaused by physical boundaries in the simulation differ-ing from the experimental apparatus, in order to makethe simulations possible.

Preliminary results for the stagnation point flame us-ing the LES-LEM approach demonstrate that the ap-plication of LES-LEM to spatially evolving flows withcomplex mean fluid flow is feasible, stationary flameposition and stationary burning rates are achieved, andthat they scale properly with U'/SL- Further work isneeded for the specification of ksgs in two-dimensionalsimulations.

AcknowledgmentsThis work was supported in part by the Air Force Of-fice of Scientific Research under the Focused ResearchInitiative, monitored by General Electric Aircraft En-gines, Cincinnati, Ohio. Support for the computationswas provided by the Army Research Laboratory, Ab-erdeen, Maryland and is gratefully acknowledged.

ReferencesAbdel-Gayed, R. G., Bradley, D., and McMahon, M.(1979) "Turbulent Flame Propagation in PremixedGases: Theory and Experiment," Stvente.e.nth Sympo-sium (International) on Combustion, The CombustionInstitute, Pittsburgh, pp. 245-254.

Abdel-Gayed, R. G., Al-Khishali, K. J. and Bradley,D. (1984) "Turbulent Burning Velocities and FlameStraining in Explosions," Proc. R. Soc. Land., A Vol.39, pp. 393-414.

Calhoon, W. H. and Menon, S. (1996) "Subgrid Mod-eling for Reacting Large Eddy Simulations," Presentedat the 34th Aerospace Sciences Meeting, Reno, Nv,January 15-18 AIAA-96-0516.

11

Calhoon, W. H. and Menon, S. (1997) "Linear-EddySubgrid Model for Reacting Large-Eddy Simulations:Heat Release Effects," Presented at the 35th AerospaceSciences Meeting, Reno, Nv, January 6-9 AIAA-97-0368.

Cho, P., Law, C. K., Hertzberg, J. R., and Cheng, R.K. (1986) "Structure and Propagation of TurbulentPremixed Flames Stabilized in a Stagnation Flow,"Twenty-first Symposium (International) on Com-bustion, The Combustion Institute, Pittsburgh, pp.1493-1499.

Cho, P., Law, C. K., Cheng, R. K., and Shepherd,I. G. (1988) "Velocity and Scalar Fields of TurbulentPremixed Flames in Stagnation Flow," Twenty-SecondSymposium (International) on Combustion, The Com-bustion Institute, Pittsburgh, pp. 739-745.

Colonius, T., Lele, S. K., and Moin, P. (1993)"Boundary Conditions for Direct Computation ofAerodynamic Sound Generation," AIAA J., Vol. 31,No. 9, pp. 1574-1582.

Erlebacher, G. Hussaini, M.Y. Speziale, C.G. andZang, T.A. (1987) "Toward the Large-Eddy Simula-tion of Compressible Turbulent Flows," Institute forComputer Applications in Science arid Engineering,Hampton, VA, ICASE 87-20.

Gottlieb, D. and Turkel, E. (1976) "Dissipative Two-Four Methods for Time-Dependent Problems," Math.ofComp., Vol. 30, No. 136, pp.703-723.

Haworth, D. C. and Poinsot, T. J. (1992) "NumericalSimulations of Lewis Number Effects in TurbulentPremixed Flames," /. Fluid Mech., Vol. 244, pp.405-436.

Herring, J. R., Orszag, S. A., Kraichnan, R. H.and Fox, D. G. (1974) "Decay of Two-DimensionalHomogeneous Turbulence," /. Fluid Mech., Vol. 66,pp. 417-444.

Kerstein, A. R. (1986) "Pair-Exchange Model ofTurbulent Premixed Flame Propagation," Twenty-firstSymposium (International) on Combustion, The Com-bustion Institute, Pittsburgh, pp. 1281-1289.

Kerstein, A.R., Ashurst, W.T., and Williams, F.A.(1988), "The Field Equation for Interface Propagation

in an Unsteady Homogeneous Flow field", Phys. Rev.A. Vol. 37, pp. 2728.

Kerstein, A. R. (1991) "Linear-Eddy Modeling ofTurbulent Transport. Part 6. Microstructure ofDiffusive Scalar Mixing Fields," J. Fluid Mech., Vol.231, pp. 361-394.

Kerstein, A. R. (1992) "Linear-Eddy Modeling of Tur-bulent Transport. Part 4. Diffusion-Flame Structureand Nitric Oxide Production," Combust. Sci. Tech.,Vol. 81, pp. 75-86.

Kim, W.-W. and Menon, S. (1996) "On the propertiesof a localized dynamic subgrid-model for large-eddysimulations," Submitted to J. Fluid Mech..

Lee, S., Lele, S. K., and Moin, P. (1992) "Simulationof spatially evolving turbulence and the applicabilityof Taylor's Hypothesis in compressible flow," Phys. ofFluids A, Vol. 4, pp. 1521-1530.

Liu, Y., and Lenze, B. (1988) "The Influence ofTurbulence on the Burning Velocity of PremixedCH^ — HI Flames with Different Laminar BurningVelocities," Twenty-Second Symposium (International)on Combustion, The Combustion Institute, Pittsburgh,pp. 747-754.

McMurtry, P. A. Gansauge, T. C., Kerstein, A. R.,and Krueger, S. K. (1993) "Linear-Eddy Simulationsof Mixing in a Homogeneous turbulent Flow," Phys.Fluids A, Vol. 5 pp. 1023-1034.

Menon, S., and Jou, W.-H. (1990) "Numerical Simu-lations of Oscillatory Cold Flows in an AxisymmetricRamjet combustor," Journal of Propulsion and Power,Vol. 6, No. 5, pp. 525-534.

Menon, S. (1991) "Active Control of Combustion In-stability in a Ramjet Using Large-Eddy Simulations,"AIAA-91-0411, 29th Aerospace Sciences Meeting,Reno, Nv, January 7-10.

Menon, S. and Kerstein, A. R. (1992) "StochasticSimulation of the Structure and Propagation Rate ofTurbulent Premixed Flames," Twenty-Fourth Sympo-sium (International) on Combustion, The CombustionInstitute, Pittsburgh, pp. 443-450.

Menon, S., McMurtry, P. A. and Kerstein, A. R.

12

(1994) "A Linear Eddy Subgrid Model for TurbulentCombustion: Application to Premixed Combustion,"Presented at the 31st Aerospace Sciences Meeting,Reno, Nv, January 11-14, AIAA-93-0107.

Moin, P., Squires, K., Cabot, W., and Lee, S. (1991)"A Dynamic Subgrid-Scale Model for CompressibleTurbulence and Scalar Transport," Phys. Fluids A,No. 3, Vol. 11.

Poinsot, T., Veynante, D. and Candel, S. (1991)"Quenching Processes and Premixed Turbulent Dia-grams," J. Fluid Mech., Vol. 228 pp. 561-606.

Poinsot, T, J., and Lele, S. K. (1992) "Bound-ary Conditions for Direct Simulation of CompressibleViscous Flows," J. Corny. Phys.,Vol. 101pp. 104-129.

Shepherd, I. G., Cheng, R. K., and Goix, P. J. (1990)"The Spatial Scalar Structure of Premixed TurbulentStagnation Point Flames," Twenty-Third Symposium(International) on Combustion, The CombustionInstitute, Pittsburgh, pp. 781-787.

Shepherd, I. G., and Ashurst, Wm. T. (1992) "FlameFront Geometry in Premixed Turbulent Flames,"Twenty- Fourth Symposium (International) on Com-bustion, The Combustion Institute, Pittsburgh, pp.485-491.

Smith, T. and Menon, S. (1996a) "The Structure ofPremixed Flames in a Spatially Evolving TurbulentFlow," Combust. Sci. Tech., Vol. 119, No. 1-6.

Smith, T. and Menon, S. (1996b) "Model Simulations ofFreely Propagating Turbulent Premixed Flames," pre-sented at the Twenty-sixth Symposium (International)on Combustion, The Combustion Institute, Pittsburgh.

Smith, T. and Menon, S. (I996c) "One-DimensionalSimulations of Freely Propagating Turbulent PremixedFlames," Submitted to Combust. Sci. Tech., Septem-ber.

Squires, K. and Zeman, 0. (1990) "On the Subgrid-Scale Modeling of Compressible Turbulence," Centerfor Turbulence Research, Proceedings of the SummerProgram.

Trouve, A. and Poinsot, T. (1994) "The EvolutionEquation for the Flame Surface Density in Turbulent

Premixed Combustion," /. Fluid Mech., Vol. 278, pp.1-31.

Yahagi, Y., Ueda, T., and Mizomoto, M. (1992) "Ex-tinction Mechanism of Lean Methane/Air TurbulentPremixed Flame in a Stagnation Point Flow," Twenty-Fourth Symposium (International) on Combustion,The Combustion Institute, Pittsburgh, pp. 537-542.

Yakhot, V. (1988) "Propagation Velocity of PremixedTurbulent Flames," Combust. Sci. Tech., Vol. 60, pp.9-24.

13

20.0

15.0 -

10.0

——— Yakhofs RNG Model (1988)O Abdel-Oayed el al. Data (1984a)

——— Abdel-Gayed ct al. Curve Fit (1984a)GLEM, CflS, L=0.178 m

•*——»LEM, Cx=15, L=0.037 m•*-—#LEM, C.=1S, L=0.178 m

0.0 L0.0 30.0

Fig. la Normalized turbulence flame speed for GLEM,and finite-rate LEM compared with Yakhot's model andthe fan stirred bomb experiments of Abdel-Gayedetal. (1984).25.0

20.0

15.0

10.0

o.o

——— Abdel-Gayed etal. (1979), Re=42:——— Abdel-Gayed et al. (1979), Re=17!———»LEM, u'=0.52 m/s, Re=488

LEM, u'=1.0 m/s, Re=939LEM, u'=1.7 m/s, Re=1594

-A LEM, u'=2.25 m/s, Re=2112

0

0.0 0.2 0.80.4 0.6SL/u'

Fig. 1 b Normalized turbulent flame speed as a function ofnormalized laminar flame speed for finite rate flames fordifferent Re compared with the fan stirred bombexperiments of Abdel-Gayed et al. (1979).

V.JVJU

0.5040

0.5030

0.5020

0.5010

0.5000 t

0.4990

0.4980

0.4970

0.4960

n /io^n

G — e U/SL=OB —— O u/SL=,5A —— Au/S,=1.5

-

• JrT^^^^-i=&^^-«=B-^F^'/

•0.0 2.0 4.0 6.0

(t/(ALES/SL))Fig. 2 One-dimensional inter-LES cell flamepropagation.

8.0 10.0

10"'

ID-

10"

— - - - Initial Theoretical Energy 1——— DNS Casel, Herring et al. \— — Filtered DNS i

- G——O LES Dynamic CoedB——El LES, No Model

10 100 1000,KFig. 3a Energy spectra for DNS and LES of decaying

isotropic turbulence taken at t=4.

——— DNS Case I, Herring et alQ——O Filtered DNSQ——BLES Dynamic Coef.-X——*LES, No Model© - -OLES Const. Coef.Q- —a Filtered DNS, k'"A——ALES Dynamic Coef., k'"H———I-LES, Const. Coef.

Fig.Sb Evolution of kinetic energy and subgrid kineticenergy.

14

(a) (b)

(c) (d)

Fig. 4 DNS of freely propagating premixed flames in homogeneous turbulence. The evolution of species mass fraction from a globalfinite-rate combustion model is compared with the G-field equation. Snapshots of vorticity and a) mass fraction at i=l, b) massfraction at t=2, c) G-field at 1=1, and d) C-field at 1=2.

15

10.0

Fig.Sa Snapshot of vorticity and filtered scalar fromLES-LEM of passive flame propagation in decayingisotropic turbulence for u'/$£=1.0 and t=2.

Fig.5b Snapshot of vorticity and filtered scalar fromLES-LEM of passive flame propagation in decayingisotropic turbulence for u'/$£=10.0 and T=2.

3.0

2.0

1.0 (

0.00.0 1.0 2.0

T

3.0 4.0

—— u'/SL=5, Re=500u'/SL=10, Re=1000

Fig.6b Evolution of normalized subgrid turbulent flamespeed.

15.0

10.0

5.0

0.0

——— u'/SL=l,Re=100——— uVSL=5, Re=500——— u'/SL=10, Re=1000

0.0 1.0 2.0T

3.0 4.0

Fig.6c Evolution of normalized average scalarconsumption rate.

Fig.6a Evolution of normalized resolved flame area.FigJa Snapshot of vorticity and G-field from cold flowstagnation point flame simulation.

16

Fig.7b. Snapshot of vorticity and G-field from reactingflow stagnation point flame simulation.

——— u , ^m, LESa—e v' Tp/T,=4, LES

u' T/I>7, LESA——A v' T/I'p?, LES•—•-• u Reacting Flow, Exp

v Reacting Flow, Exp

0.04 0.05 0.070.06X[m]

Fig.Sc R.M.S. turbulence intensities for reacting flowstagnation point flames.

0.08

0.045 0.050 0.055 0.060 0.065 0.070 0.075X[m]

Fig.Sa Mean axial flame locations.

0.8

0.6

——— u Exp. 'Cho et al. (Q——0 v' Exp. Cho et al. (1986)•——• u LESA——A v LES

0.04 0.08

1500750

I -750•-1500

Fig.9a Snapshot of vorticity and the filtered scalar fieldfrom LES-LEM stagnation point flame.

40.0

30.0

6E

20.0

10.0

0.0

Q——Q Subgritl ti/SL, u'/SL-:B——DS0/SU, uVSL=2Q——OA/AL. u'/SL=lG——CSubEridu/SL. u'/SL=

S^ UVSL=1

4.0 5.0 6.0 7.0

Fig.Sb R.M.S. turbulence intensities for cold flowstagnation point flames.

Fig.9b Evolution of propagation rates from LES-LEMsimulation of stagnation point flame.

17

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