1995

Proceedings of the Combustion Institute, Volume 29, 2002/pp. 1995–2000

LARGE EDDY SIMULATION OF A PREMIXED FLAME WITH APPROXIMATEDECONVOLUTION MODELING

JOSEPH MATHEWDepartment of Aerospace Engineering

Indian Institute of ScienceBangalore 560 012, India

Approximate deconvolution modeling is a very recent approach to large eddy simulation of turbulentflows. It has been applied to compressible flows with success. Here, a premixed flame which forms in thewake of a flameholder has been selected to examine the subgrid-scale modeling of reaction rate by thisnew method because a previous plane two-dimensional simulation of this wake flame, using a wrinklingfunction and artificial flame thickening, had revealed discrepancies when compared with experiment. Thepresent simulation is of the temporal evolution of a round wakelike flow at two Reynolds numbers, Re �

2000 and 10,000, based on wake defect velocity and wake diameter. A Fourier-spectral code has beenused. The reaction is single-step and irreversible, and the rate follows an Arrhenius law. The referencesimulation at the lower Reynolds number is fully resolved. At Re � 10,000, subgrid-scale contributionsare significant. It was found that subgrid-scale modeling in the present simulation agrees more closely withunresolved subgrid-scale effects observed in experiment. Specifically, the highest contributions appearedin thin folded regions created by vortex convection. The wrinkling function approach had not selectedsubgrid-scale effects in these regions.

Introduction

Large eddy simulation (LES) is well-establishedfor incompressible flows. The dynamic subgrid-scalemodel [1], which continuously estimates a model pa-rameter during the course of the simulation, hasproved to be the adequate development to take LESto analyses of incompressible, engineering flows [2].This success has encouraged searches for methodsto make LES more widely applicable. Turbulent re-acting flows present both the need and the oppor-tunity for applications of LES. Such flows exhibitlarge-scale organization which determines mixingand, in turn, the progress of reactions. The associ-ated large-scale unsteadiness places severe demandson modeling for direct computations of the meanfield (Reynolds-averaged Navier-Stokes simula-tions). LES is a natural choice for such flows becausethere is no modeling of these large-scale structures.But, LES for reacting flow is still developing be-cause, in addition to special difficulties, the heuristicmodeling allowed by incompressible flow cannot beextended to general flows.

LES requires models for the effects of unresolvedsmall scales on the large scales arising from everynonlinear term. These are often termed subgrid-scale (SGS) models denoting the modeling of scaleswhich are smaller than grid spacing. For incom-pressible flows, the nonlinear terms are the convec-tion terms, and SGS effects appear as a tensor, likethe stress tensor. Heuristic SGS modeling, extended

from physical models, takes this tensor to depend onthe computed strain rate similar to the models forReynolds stress. In both cases, the basic physicalmodel is the stress-strain rate relationship for New-tonian fluids. For general flows, like compressible orreacting flows, there are other nonlinear terms.Then, modeling must be done in some other waybecause physical models to guide heuristic modelingmay not be available for every term. Often, severalterms are either ignored or assumed to be modeledtogether. A recent development in LES which cir-cumvents this difficulty is the approximate decon-volution model (ADM) proposed by Adams andLeonard [3]. Several papers reporting excellent re-sults from a priori and a posteriori tests of LES withADM for non-reacting, compressible flows have ap-peared [4–6]. The present study was undertaken toexamine the effectiveness of ADM for reacting flow.Other simulations of compressible flows have al-ready shown that ADM is able to provide good mod-eling for convection. So, the specific nonlinearityconsidered here is the Arrhenius reaction rate term.To add focus, a premixed flame which forms in thewake of a flameholder has been considered becausean LES of this configuration, studied by Nottin etal. [7] using other techniques, was reportedly foundwanting: there were qualitative differences in theSGS contributions between laboratory measure-ments and LES computations. They had used a Sma-gorinsky-type model for the flow and a wrinklingfunction for the reaction rate term. They concluded

1996 TURBULENT COMBUSTION—Large Eddy and Direct Numerical Simulations

that the sensor for the wrinkling function was inade-quate and also that some kind of dynamic procedurewas needed. The ADM is potentially better becauseit combines beneficial effects of the Smagorinskyand scale similarity models and is also a dynamicprocedure.

LES of realistic premixed flames faces an essentialdifficulty: The flame thickness is much smaller thanthe smallest turbulence eddy scale we wish to resolvein an LES. One approach, in different manifesta-tions, has been to track the flame front (G equation)or find the flame surface density [8,9] or the flamewrinkling factor [10] and thence obtain the low-passfiltered reaction rate. Another approach has been toobtain a partial resolution of flame structure on theLES grid by thickening the flame artificially and alsouse a wrinkling function to compensate for the mod-ified flame-turbulence interactions of the thickenedflame. Nottin et al. [7] used the thickened flamemethod to simulate an experiment in which a pro-pane/air mixture burns in the wake of a flame holderwith upstream acoustic excitation. A two-dimen-sional simulation was considered adequate for theflow near the flame holder. They found global statis-tical quantities from simulation and experiment toagree very well, but local quantities showed discrep-ancies. Specifically, the SGS contributions to reac-tion rate in the simulation were largest in stretchedregions around shear layer vortices, whereas themaximum of the unresolved part in experiments oc-curred in the folds of concentration between a vortexand braid.

Although ADM may offer benefits when incor-porated into other approaches to reacting flow LES,the present study has the limited aim of assessing itspotential for the thickened flame method. A flowwhich is qualitatively similar to that considered byNottin et al. [7] has been simulated to determinewhether the discrepancy which they observed canbe overcome. ADM is described in the followingsection. Characteristics of the underlying direct nu-merical simulation (DNS) code have been discussedpreviously. Accordingly, only the specifics for pre-mixed reacting flow are presented. Next, overallcharacteristics of the simulations at two Reynoldsnumbers (fully resolved at 2000 and requiring ADMat 10,000) are presented. A detailed examinationthen shows that ADM provides SGS contributionsto the reaction rate which are closer to experiments.

Approximate Deconvolution Modeling

The essential aspects of this method can be un-derstood by applying it to the generic, one-dimen-sional transport equation for u(x,t),

�u �f(u)� � 0 (1)

�t �x

where f(u) is a nonlinear function. Computing a

large wavenumber solution, as in an LES, implies afiltering of equation 1:

�u �f(u)� G * � 0 (2)

�t �x

Here, G is a low-pass filter and filtered quantitiesare obtained by convolution: u � G * u � � G(x � x�) u(x�) dx�. The nonlinear term can also bewritten in terms of the filtered variable. Then equa-tion 2 reads

�u �f(u)� � R (3)

�t �x

with a remainder

�f(u) �f(u)R � � G *�x �x

For an LES of incompressible flow, the differentialequation 1 is replaced by the Navier-Stokes equa-tions, and the remainder is the SGS stress term,��sij/�xj, where sij � � uiuj and ui are velocityu ui jcomponents.

Subgrid-scale modeling finds models Rm(u) forthe remainder as a function of the computed solu-tion u. While eddy viscosity models replace this termwith a strain rate, the scale similarity model of Bar-dina [11] uses the computed field to estimate thisterm. The approximate deconvolution model of Ad-ams and Leonard [3] sets

�f(u) �f(u*)R � � G * (4)m �x �x

where u*(x,t) � Q * u is an approximation to u(x,t)obtained by approximate deconvolution. On substi-tuting equation 4 into equation 3, and assuming,consistently with equation 4, that G * u* � G * u,the LES problem reads

�u* �f(u*)G * � � 0 (5)� ��t �x

For a numerical solution, equation 5 can be split intoan integration of the original differential equation 1without any explicit SGS term, followed by a filter-ing. For example, with superscript n indicating time-stepping, we can find u*(n) � Q * u(n) by approxi-mate deconvolution of the LES field u(n), advancethrough an integration step

�u*(n�1) (n) 2u* � u* � Dt � O(Dt) (6)�t

and filter to obtain the LES field u(n�1) � G *u*(n�1). This three-stage procedure ensures that theSGS modeling prescribed by the ADM (equation 4)has been incorporated. When executed as a se-quence, the two filtering steps between each inte-gration step (equation 6) are equivalent to the single

LES WITH APPROXIMATE DECONVOLUTION 1997

filtering Q * G * If Q were the exact inverse(n�1)u* .of G, the resultant filtering Q * G * would(n�1)u*not alter the field. But, as an approximate inverse,QG removes content at the high wave number end.This equivalence between filtering and the approx-imate deconvolution model implies that implemen-tation in DNS codes involves just an explicit filteringstep.

In addition to the ADM procedure discussedabove, Adams and Leonard [3] added an ad hoc reg-ularization term v(G2 � I) * u to their filtered equa-tions. They took G2 � QG. It is straightforward toshow that the effect of adding this term is equivalentto filtering u with G2 every 1/(vDt) time steps withinthe temporal truncation error of the integrationmethod. Such an interpretation allows the use of ex-isting DNS codes easily: integration procedures arenot modified. Instead, there is an additional, or sec-ondary, filtering of the variables between time-steps.It has also been shown by Mathew [12] that the formof the ad hoc term is close to that which can beobtained by a formal procedure leading to suchterms for every nonlinear term. But, using the adhoc term is simpler and seems to have been ade-quate.

Reacting Flow Modeling

The flow is governed by the Navier-Stokes equa-tions. For premixed flames at unit Lewis number, aprogress variable c(x, t) determines species concen-trations (mass fractions Yi; i � F, O, P for fuel, ox-idizer, and product) and temperature T from the re-lations: c � YP/YP,b � (T � Tu)/(Tb � Tu), YF �(1 � YP)/(1 � c), and YO � cYF. Subscripts b andu denote burned and unburned states, respectively.The reaction is single-step and irreversible. For sim-plicity, the unburned mixture was taken to be at thestoichiometric ratio; c � 1 and YP,b � 1. The trans-port equation for the progress variable is

1 2Dc � � c � w (7)PReSc

where D � �/�t � u • � and u is the velocity. Twovalues of the Reynolds number Re were used, butthe Schmidt number Sc � 1 in both simulations.The Arrhenius type reaction rate model consideredhere is wP � AYFYO exp(�Ta/T), and wP � wP(c)only. In the simulations, A � 105, Ta/Tb � 8, andTb/Tu � 2.

On filtering equation 7 to perform an LES, re-mainder terms of the form introduced in equation 3appear. For the convection and reaction rate terms,they are Rt � u • �c � and Rw �u • �c

� wP(c), respectively. These are the subgridw (c)Pcontributions which were obtained from ADM.

Simulations and Results

The experiment considered by Nottin et al. [7] isof a premixed flame which is held in the wake of atriangular prism flameholder in a rectangular duct.Their simulation was a plane two-dimensional ap-proximation of the experiment. The flow was acous-tically forced using loudspeakers placed upstream.The present simulations are of the temporal evolu-tion of unforced, round, wake-like flows. Flow evo-lution at successive times in the simulation can becompared to flow development at successive down-stream stations of laboratory flows. A notable ap-proximation is that these are incompressible flowsimulations, but it should be acceptable because thestudy is of the modeling of the Arrhenius reactionrate term. Also, compressibility effects were not im-portant for Nottin et al. [7] either in their low Machnumber flow. However, significantly, the presentthree-dimensional simulations preserve essentialturbulence dynamics.

Simulations were performed using a code whichhad been developed for DNS. Characteristics of thealgorithm and DNS solutions have been reportedbefore [13,14]. It is a Fourier-spectral algorithm inCartesian coordinates with physical space evaluationof nonlinear terms. The initial wake-like velocityprofile was constructed from a thin cylindrical vortexsheet. The streamwise component rises from zero inthe cylindrical core to a level across the vortex sheet(tanh velocity profile). The velocity scale was U andthe mean diameter D of the initial vortex sheet wasthe length scale. The Reynolds number Re is basedon these two scales. The initial flame lies in the an-nular shear layer enclosing product and surroundedby premixed reactant. The product distribution (YP �1 � u) mimics the region immediately downstreamof flame holders where burned gases accumulate.

Two simulations are presented below. The initialfields were the same. Both solutions had periods3D � 3D � 3D on a grid of 64 � 64 � 64 points.The first simulation is at Re � 2000. At this relativelylow Reynolds number, the flow is essentially laminarand there is no loss of resolution on this grid. Sub-grid-scale modeling is not required. The second isat Re � 10,000. Now, the initial evolution throughvortex ring dynamics is faster and is followed by abreakdown to turbulent flow. ADM provides SGSmodeling. The filter G (equation 2) was a Gaussian-like, Pade-type filter specified by a parameter �0.5� � � 0.5. When � is increased, filter width in-creases and the cutoff of the effective filter QGmoves closer to the high wave number end (see Ad-ams and Leonard [3] for details). Here, a high value� � 0.4 could be used because the underlyingscheme is spectral.

The initial vortex sheet suffers a Kelvin-Helm-holtz-like instability. Vorticity grows and rolls up intoa sequence of vortex rings. Since the most amplified

1998 TURBULENT COMBUSTION—Large Eddy and Direct Numerical Simulations

Fig. 1. Distributions on sections from simulation atRe � 2000 (t � 5). Gray level varies from light to dark asthe value of the quantity increases. (a) Vorticity magnitudeon longitudinal section |� � u|; (b) product mass fractionYP � c.

Fig. 2. Distributions of reaction rate wP(c) at successivetimes (t � 2,3,4,5,7, and 9 from left to right). One-half ofstreamwise period is shown in each frame. Gray level variesfrom light to dark as reaction rate increases. (a) Re � 2000;(b) Re � 10,000.

wavelength scales with sheet thickness, small per-turbations were added to select this rollup and a sub-sequent pairing [13]. As an example of the solution,Fig. 1a shows vorticity distribution on a longitudinalsection at t � 5 (Re � 2000). A sequence of vortexrings has formed. Product distribution is shown inFig. 1b. The action of the vortex rings bringing insurrounding fluid can be seen as tongues of lowerconcentration along braids between neighboringrings. Reaction rates (part of Fig. 2a, discussed be-low) are highest around ring vortices and in thebraids where unburned fluid has reached and tem-peratures are high.

In temporal simulations, flow structures whichform at successive times correspond to those whichdevelop at successive downstream stations in labo-ratory wakes. For a comprehensive picture, reactionrate fields at successive times (t � 2,3,4,5,7,9) have

been arranged in a sequence in Fig. 2. One-half ofthe streamwise extent at each time has been shown.Fig. 2a is for Re � 2000 and Fig. 2b is for Re �10,000. Although the initial evolution looks similar,soon differences appear. At the lower Reynoldsnumber the reaction zone eventually becomes anearly uniform annular sheet, insensitive to the vor-tices which dissipate. At the higher Reynolds num-ber, the reaction zone fills the wake due to strong(turbulent) mixing. The total mass of product scaledwith the initial mass is plotted as a function of timein Fig. 3. The greater mixing in a turbulent flow canbe seen in the significantly higher product formationrate following breakdown of the flow beyond t � 7.

The discrepancies in SGS contributions observedby of Nottin et al. [7] were close to neighboring vor-tices. Fig. 2 shows that the data at t � 4 (Re �10,000), when a sequence of large vortices with well-defined braids exist, can be used to examine this is-sue in the present simulation. Fig. 4 shows distri-butions of product, filtered reaction rate w (c*),Pvorticity, and the SGS reaction rate � wP(c).w (c*)PFour parts of a vortex have been marked in Fig. 4d.We may term A the outer rear, B the outer head, Cthe inner fold, and D the braid. In experiments, Not-tin et al. [7] found unresolved SGS values to be

LES WITH APPROXIMATE DECONVOLUTION 1999

Fig. 3. Growth of mass of product scaled with initialmass. (—) Re � 2000; (—) Re � 10,000.

Fig. 4. Distributions on sections from simulation atRe � 10,000 (t � 4). (a) Product mass fraction YP � c;(b) filtered reaction rate (c) vorticity magnitude onw (c*).P

longitudinal section |� � u|; (d) subgrid-scale contributionto reaction rate � wP(c). Quantities in (a)–(c) arew (c*)P

positive, while SGS term in (d) takes on positive and neg-ative values. Gray level is darker for larger magnitudes.

strongest in region C where the flame is folded closetogether and has high curvature. Subgrid contribu-tions from the present ADM simulations are alsohighest in region C. The reaction rate is also highesthere. Fig. 4b shows reaction rate reach peak levelswhere fresh reactants have been carried into the thinfolds between vortex and braid (see also Fig. 4a).Subgrid values are also significant in region A, both

in their experiments and here. But, in the experi-ments there was little contribution in the braids (D)unlike here. In their simulations [7], subgrid valueswere highest in region B, unlike both their experi-ment and the present simulation. High levels wereseen in regions A and D but not in C. So we findthat ADM is closer to experiment with regard to thefolded regions, but differs in the contributions at thebraids. Also, unlike the wrinkling function which de-termined from the vorticity field, here the SGS mod-eling in ADM depends directly on the term it orig-inates from: The vorticity has peaks within the ringvortex cores in Fig. 4c but the SGS effect on reactionrate vanishes in the cores.

It must be noted that there were other differencesbetween simulation and experiment: In experi-ments, reaction zones had become thicker and peaklevels had diminished with downstream distance. Intheir simulations, reaction rate intensity persisteddownstream, and the flame remained thinner. Theabsence of SGS effects in the braids in experimentcould be due to this thickening. Then, the reactionrate can be resolved and SGS effects disappear. Thisargument is further supported by the presence ofSGS effects in the thin, first braid (next to flameholder; see Fig. 4d in Ref. [7]). So SGS effects seenin the braids in the present ADM simulation andtheirs need not be a discrepancy in modeling; rather,the cause is a difference in underlying flow structure.Taken together, there is strong support that ADMselects the right SGS contributions to reaction rate.

Conclusions

An LES of a premixed flame forming in the shearlayer of a round wake has been performed using theapproximate deconvolution model. Other simula-tions of compressible flows have already shown thatADM is able to provide good modeling for convec-tion. A close examination of the SGS effects pro-vided by ADM for an Arrhenius reaction rate termshows that it is close to that observed as unresolvedsmall-scale effects in experiments of a similar flow.This is in contrast to using the wrinkling functionwith a thickened flame. It remains to be determinedwhether artificial thickening is the best-suited forLES of reacting flow but present results suggest thatADM may be the better complement to the thick-ened flame method.

Acknowledgments

It is a pleasure to thank Prof. R. Friedrich of TU-Munichwho introduced me to the ADM for LES. I also thank Dr.Amit J. Basu who developed the underlying DNS code.

REFERENCES

1. Germano, M., Piomelli, U., Moin, P., and Cabot,W. H., Phys. Fluids A 3(7):1760–1771 (1991).

2000 TURBULENT COMBUSTION—Large Eddy and Direct Numerical Simulations

2. Moin, P., in Turbulence and Shear Flow Phenomena-2(E. Lindberg et al., ed.), Royal Institute of Technology,Stockholm, 2001, pp. 1–10.

3. Adams, N. A., and Leonard, A., in Direct and Large-Eddy Simulation III (P. Voke et al., ed.), Kluwer Ac-ademic Publishers, Dordrecht, 1999, pp. 201–212.

4. Stolz, S., and Adams, N. A., Phys. Fluids 11:1699–1701(1999).

5. Stolz, S., Adams, N. A., and Kleiser, L., Phys. Fluids13:997–1015 (2001).

6. Stolz, S., Adams, N. A., and Kleiser, L., Phys. Fluids13:2985 (2001).

7. Nottin, C., Knikker, R., Boger, M., and Veynante, D.,Proc. Combust. Inst. 28:67–73 (2000).

8. Boger, M., Veynante, D., Boughanem, H., and Trouve,A., Proc. Combust. Inst. 27:917–925 (1998).

9. Hawkes, E. R., and Cant, R. S., Proc. Combust. Inst.28:51–58 (2000).

10. Weller, H. G., Tabor, G., Gosman, A. D., and Fureby,C., Proc. Combust. Inst. 27:899–907 (1998).

11. Bardina, J., ‘‘Improved Turbulence Models Based onLarge Eddy Simulations of Homogeneous, Incom-pressible, Turbulent Flows,’’ Ph.D. thesis, Departmentof Mechanical Engineering, Stanford University, Stan-ford, CA, 1983.

12. Mathew, J., The Deconvolution Method for Large EddySimulation, report 2000-FM-7, Department of Aero-space Engineering, Indian Institute of Science, Ban-galore, 2000.

13. Mathew, J., and Basu, A. J., Flow Turbulence Combust.64:71–93 (2000).

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COMMENTS

Stephen B. Pope, Cornell University, USA. There is asubstantial difference between the subgrid modeling of thestress and the reaction rate. For the stress, the dominantcontribution arises from motions that are just smaller thanthe smallest resolved scales, whereas for the reaction rate,the contributions may arise from much smaller scales. Canthe ADM procedure accurately represent the subgrid re-action rate in such circumstances?

Author’s Reply. ADM is a mathematically consistenttreatment of small scales close to the smallest representedscales in a calculation. For ADM to work satisfactorily, un-representative small scales must have a small effect on thesolution. Smaller scales must not be an essential part of theproblem. These are general expectations in LES. There-fore, ADM can be useful for the reaction rate for thick, orthickened, flames. When flame structure is thin, other suit-able modeling will be required for the reaction rate termeven when ADM is adequate for other nonlinearities. Thisis analogous to wall layer treatments in LES that eitherresolve wall layer structure with a fine grid or use an ad-ditional model.

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Sebastien Candel, Ecole Centrale Paris, France. Youhave nicely illustrated the ADM method. However, the cal-culations are rather far from conditions prevailing in theexperiment. In the case of a ducted flame the fresh gasesare initially moving faster than the burned reactants butfurther downstream the situation is reversed because the

hot products accelerate. This strongly influences the shearlayer development.

Author’s Reply. A model flow whose flow structure re-sembles the near wake flow in the experiment was chosen.The temporal, constant density simulation was not ex-pected to closely model a ducted flame. It was pointed outthat in the present ADM simulation, SGS contributions tothe reaction rate appeared in the folded regions as well. Awrinkling function may still be needed as an adjunct toADM for quantitatively close results, and such a quanti-tative assessment must use a close model of the flame.

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Anaias Tomboulides, ETH Zurich, Switzerland. In theADM procedure, there is a relaxation term accounting forenergy transfer from large to small scales. It would be in-teresting to learn how you dealt with that term, particularlyin the species concentration equation that includes the re-action rate.

Author’s Reply. The ad hoc term which is used withADM and its interpretation as a secondary filtering hasbeen discussed in the paper. This is higher-order modeling,and its effectiveness has not been established yet. Primaryand secondary filtering result in a single equivalent filter,and the solution depends on this equivalent filter only andnot separately on the two constituents. The present simu-lations were conducted with a single equivalent filter, witha very high cutoff wave number, for all variables includingconcentration.