Application of a flame-wrinkling les combustion model to a turbulent mixing layer

Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 899–907

APPLICATION OF A FLAME-WRINKLING LES COMBUSTION MODEL TO A TURBULENT MIXING LAYER H. G. WELLER,1 G. TABOR,1 A. D. GOSMAN1 and C. FUREBY2 1Department of Mechanical Engineering Imperial College of Science, Technology and Medicine London SW7 2BX, UK 2National Defence Research Establishment (FOA) S-172 00 Stockholm, Sweden

The necessity for turbulent combustion modeling in the large-eddy simulation (LES) of premixed turbulent combustion is evident from the computational cost and the complexity of handling flame kinetics reaction mechanisms directly. In this paper, a new flame-wrinkling LES combustion model using conditional filtering is proposed. The model represents an alternative approach to the traditional flame-surface density based models in that the flame distribution is represented by a flame-wrinkle density function and that the effects of flame stretch and curvature are handled through a modeled transport equation for the perturbed laminar flame speed. For the purpose of validating the LES combustion model, LESs of isothermal and reacting shear layers formed at a rearward-facing step are carried out, and the results are compared with experimental data. For the isothermal case, the agreement between LES and the experimental data is excellent. For the reacting case, the evolution and topology of coherent structures is examined, and direct comparisons are made with time-averaged profiles of velocity and its fluctuations, temperature, and reaction products. Good agreement is obtained, to a large extent due to accurate modeling of the flame-wrinkle density but also to the novel treatment of the strain-rate effects on the laminar flame speed of the lean propane-air mixture.

Introduction Combustion processes play an important role in a wide variety of industrial applications. In many of these, the chemical reaction zone exists in a turbulent fluid dynamical environment comprising unsteady motions characterised by a wide range of length and timescales, in some cases in complex time-varying geometries. Turbulent premixed flames are important because of their occurrence in spark ignition engines [1] and gas turbines [2]; they are also fundamental to our understanding of more complicated combustion phenomena [3]. Their behavior is difficult to describe because several interconnected processes—reaction, diffusion, and volume expansion—occur in inhomogeneous flows. The challenges facing designers of combustion devices involve scale-dependent dynamic behaviors that cannot be simulated well with standard ensemble or time-averaged flow models, and more accurate methods are required. An alternative to conventional single-point Reynolds average flow models of turbulent reacting flows is large-eddy simulation (LES), which involves direct numerical simulation of the large scales and modeling of the small scales. While LES is now routinely used in simulations of nonreacting flows, its application to reacting flows is currently being developed.

Recently, Grinstein et al. [4] applied monotone integrated LES to a reacting square jet; a linear eddy subgrid combustion model is under development by Menon and Kerstein (e.g., Ref. [5]), while Mo¨ller et al. [6] applied different types of subgrid combustion models to a bluff-body stabilized flame; and Piana et al. [7] studied the G-equation approach with particular application to a plane flame. The present study is concerned with the development, application, and assessment of an LES flame-wrinkling combustion model, derived using conditional filtering. An outline of this model is given in the next section. A validation study of a reacting shear layer formed at a rearward-facing step, using experimental data, is then presented.

A New Flame-Wrinkling LES Combustion Model A full mathematical model of turbulent combustion comprises the conservation of mass, momentum, energy, and mass fractions of all constituents of the reaction mechanism [8]. Applying a densityweighted, localized, spatial filter to the governing equations yields the LES equations [9]. To close these equations, models for the subgrid scale (SGS) stress tensor, flux vectors, and dissipation as well as

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for the filtered reaction rates are required. The SGS stress tensor and flux vectors are not unique to reacting flows, and standard models may be applied; throughout this study, a one-equation eddy-viscosity model is used [10]. The principle difficulty in reacting LES is the proper treatment of the reaction zone; since the characteristic scales for the reaction processes are below the filter width, SGS reaction rate models are required. In this study, the laminar-flamelet approach is used with conditional filtering to create a set of transport equations representing the complex combustion process. Such an approach has already proved successful in the Reynolds averaged simulations (RANS) for a range of combustion problems [11–13]. By conditioning the continuity equation on the unburned gas state before filtering, a transport equation for the resolved part of the unburned gas ˜ mass fraction or regress variable b, ]qb ¯˜ ˜ 4 1q¯ S N|¹b| ¯ ` ¹s•(qU ¯ ˜ ub) u u ]t

(1)

is obtained, where the overbar symbols ¯ and ˜ represent the filtering and density-weighted filtering operations, respectively, the subscripts u and b indicate conditioning on the unburned and burned gases, respectively, the resolved unburned gas volume fraction b¯ is related to b˜ through q¯ ub¯ 4 qb, ¯ ˜ and N is the subgrid flame wrinkling. N can be regarded as the turbulent to laminar flame speed ratio and is formally related to the flame area density by R 4 ¯ N|¹b|. The conditionally filtered unburned gas velocity ˜ u is modeled using U ˜u4U ˜ ` (1 1 b˜ )U ¯ ub, where U ¯ ub 4 U ¯u1 the unburned–burned gas slip velocity U ¯ b. An algebraic model for U ¯ ub is proposed in Ref. U [11], which allows for both gradient and countergradient transport (considered important in most premixed and partially premixed combustion devices) and has proved successful in many RANS calculations [11–13]. This model may also be used in LES in the form ¯ ub 4 U

1q¯

q¯ u b

2

1 1 SuNnˆ 1 D

¹b˜ ˜ 1 b) ˜ b(1

(2)

where q¯ u/q¯ b is the density ratio between the unburned and burned states, D is the subgrid diffusion coefficient, Su is the laminar flame speed, and resolved flame normal nˆ 4 ¹b˜ /|¹b|. Combining equations (1 and 2) yields ]qb ¯˜ ˜ 4 1q¯ S N|¹b| ˜ ` ¹ • (qUb) ¯ ˜ ˜ 1 ¹ • (qD¹b) ¯ u u ]t (3) thus conveniently including the countergradient transport modeling from equation 2 in the burningrate term on the right-hand side of equation 3.

From the transport equations for the subgrid flame area density R and the resolved unburned gas ¯ a transport equation for the suvolume fraction b, bgrid flame wrinkling N is obtained [11]:

} } ]N ` U•¹N 4 1n•(¹Us)•nN ]t } } } ¹|¹b| ¯ ` nˆ•(¹U t)•nˆN ` (U t 1 U s)• N ¯ |¹b|

(4)

} where U t is the surface-filtered}effective velocity of the flame defined by ]b¯ /]t ` U t¹b¯ 4 0 and Us is the local instantaneous velocity of the flame surface. The first and second terms on the right-hand side of equation 4 represent the effects of strain and propagation (both resolved and subgrid) on N. These are modeled by decomposition}into the}resolved strain rates rt and rs (relating to U t and U s, respectively) and the subgrid turbulent generation and removal rates GN and R(N 1 1), where the rate coefficients G and R require modeling. The third term on the right-hand side of equation 4 represents the effect of differential propagation on the distribution of N through the flame, reducing generation at the front of the flame and enhancing generation at the back. This term involves high-order derivatives that create numerical difficulties for LES and are avoided by including the effect directly into the model for G, resulting in the simplified equation for N } ]N ` U s•¹N 4 GN 1 R(N 1 1) ]t ` (rs 1 rt)N

(5)

A spectral approach is applied to the modeling of the turbulence–flame interaction in which the wrinkling of the flame is decomposed into a length-scale spectrum [14]. This approach lends itself naturally to LES in that subgrid flame properties may be obtained by integrating over the appropriate range of the spectrum. Solution of the spectral evolution equations coupled with the transport equation for N is prohibitively expensive and simple algebraic models are considered more appropriate. The current approach is based on the flame-speed correlation of Gu¨lder [15], which has proved particularly good by comparison with full spectral solutions, leading to G4 R

Neq 1 1 0.28 N*eq , R4 Neq sg N*eq 1 1

!S

N* eq 4 1 ` 0.62

u8

(6)

Rg,

u

Neq 4 1 ` 2(1 1 b)(N* eq 1 1)

(7)

where sg is the Kolmogorov timescale, u8 is the subgrid turbulence intensity, and Rg is the Kolmogorov

A FLAME-WRINKLING LES COMBUSTION MODEL

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Reynolds number. The surface-filtered velocity of } the flame U s is modeled in a similar manner to the conditionally filtered unburned gas velocity as

resolved strain rate (subgrid strain and curvature effects are assumed to largely cancel as suggested by the spectral model), and assuming linear response results in

} ˜ ¯ ˜ ` (q¯ u/q¯ 1 1)SuNn˜ 1 ¹•(qD¹b) Us 4 U n˜ ˜ q|¹b| ¯ (8) ˜ The resolved strain rate rt is obtained from U and the subgrid turbulent flame speed SuN by removing the dilatational component from the strain rate in the direction of propagation nˆ resulting in

S`u 4 S0u max(1 1 rs/rext, 0)

˜ ` SuNn˜) rt 4 ¹•(U ˜ ` SuNn˜)]•n˜ 1 n˜•[¹(U

(9)

Note that in this model, the gas expansion due to combustion is assumed to occur only in the direction nˆ. This approach avoids the need to accurately model the surface-filtered gas velocity that has proved difficult. The surface-filtered resolved strain rate rs is obtained similarly except that the effects of flow-field strain and propagation strain (due to the propagation of the curved flame front) are separated in order that the influence of flame wrinkling may be modeled appropriately; in the limit of very high wrinkling (assuming isotropy), the compressive and extensive effects of the flow field are assumed to cancel, whereas the effect of propagation strain approaches half that of the wrinkling aligned with nˆ. The result is rs 4

˜ 1 nˆ•(¹U)•n ˜ ˜ ¹•U N (N ` 1){¹•(Sun˜) 1 nˆ•[¹(Sun˜)]•n˜} ` 2N

(10)

During lateral compression, it is expected that the resolved strain rate will cause an increase in flame wrinkling; however, spectral modeling [14] suggests that there is an overall slight removal rate caused by a reduction in the inter-wrinkle distance that increases the subgrid removal rate disproportionately. To include this effect (pending a more detailed investigation), a “max” function is currently applied to the last term in equation 5 allowing only removal of wrinkling by resolved extensive strain. For some combustion problems, it is reasonable to assume the laminar flame speed is unaffected by strain and curvature in which case the unstrained value from experiment and correlation may be used; however, for most, strain effects cannot be ignored [16]. Indeed, for the current problem, it has been found that the effect of the strain near the step in reducing the effective reaction rate is critical for the formation of the Kelvin–Helmholtz instability. One possible approach is to assume that the laminar flame speed is in local equilibrium with the local

(11)

where rext is the strain rate at extinction (obtained from the Markstein length by extrapolating to Su → 0). Unfortunately, the chemical timescales of lean flames may be comparable to the strain and transport timescales, in which case local equilibrium may not be assumed and a full transport equation is required. By analogy with the transport of the flame wrinkling, the filtered laminar flame speed is expected to be transported at the surface-filtered ve} locity of the flame U s, and assuming that the strainrate timescale is 1/rs and the chemical timescale is such that at as t → `, Su → S`u, then } (S0u 1 Su) ` U s•¹Su 4 1rsSu ` rsS`u ]t (Su 1 S`u) (12)

]Su

This transport equation obeys the important constraints on the laminar flame speed and has a consistent form with the other equations of the LES combustion model. However, a rigorous validation is not possible until appropriate data on the effect of time-varying strain on laminar flames becomes available. The proposed flame-wrinkling model can be simplified by the replacement of the N equation 5 with the equilibrium expression equation 7 and further by the replacement of the Su equation 12 with the equilibrium expression equation 11. These models will be referred to as the two-equation, one-equation, and algebraic models, respectively. The LES equations are discretized using an unstructured finite-volume method [17]. Second-order schemes are used in space and time, central differencing for velocity, a bounded NVD scheme for scalars [17], and three-point backward differencing in time. To decouple the pressure-velocity system, a Poisson equation, derived from discretized versions of the continuity and momentum equations, is constructed for the pressure. In the current formulation, a low-Mach number approximation is used, eliminating acoustic waves from the model (found experimentally to be unimportant for this case [18]). The set of scalar equations are solved sequentially with iteration over the explicit coupling terms to obtain convergence. Such a segregated approach results in a Courant number restriction; it is found that a Courant number of 0.5 gives satisfactory numerical stability, but a value of 0.2 is preferable for temporal accuracy.

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Fig. 1. (a) Schematic of the computational domain (h 4 0.0254 m) together with (b) an instantaneous representation of the nonreacting flow in terms of the iso surface U¯1 4 0 separating the shear layer from the recirculation region, and contours of the spanwise vorticity x ¯ 3 at the center plane.

LES of a Reacting Shear Layer at a RearwardFacing Step The case under consideration is a premixed propane-air flame with an equivalence ratio of f 4 0.57, stabilized in a turbulent-free shear layer formed at a rearward-facing step (Fig. 1a). The mean velocity, pressure, and temperature at the inlet are u0 4 13.3 m/s, p0 4 1 atm, and T0 4 293 K, respectively, resulting in a bulk Re number (based on u0 and the step height h) of Re 4 22,100. The experimental configuration [19] consists of a rectilinear premixing section followed by a smooth contraction to one-half of its height, a step expansion into the test section, and a quenching water spray-filled converging exit region. In the simulations, the smooth contraction region is omitted, and no attempt is made at representing the effects of the spray; however, the converging exit region is present in order that the outlet boundary condition be sufficiently far from the recirculation zone. All simulations start from rest, and the unsteady flow characteristics evolve naturally. The 320,000 cell grid is concentrated in the shear-layer region and near the channel walls and graded in other regions. Refinement shows that the resolution is adequate; using more cells resolves further details, but the differences in averaged quantities are small. At the inlet, Dirichlet conditions are used for all variables except p¯, for which a zero Neumann condition is applied. The inlet velocity is obtained from the experimentally measured mean profile, on which “white noise” of 4%, 2%, and 2% is superimposed in the streamwise, spanwise, and normal directions, respectively. Although the application of “white noise” is a poor representation of turbulence, it has proved adequate for the current

case, with the results found to be insensitive to the level of the prescribed fluctuations. The inlet SGS k is taken as the measured turbulence kinetic energy. At the outlet, a transmissive condition is used for p¯, and all remaining variables are subjected to zero Neumann conditions. A no-slip condition is applied to the velocity at the walls in conjunction with a vanDriest damping function, while all scalar-valued quantities are subjected to zero Neumann conditions. Periodic conditions are applied in the spanwise direction. Data for the laminar flame speed and extinction strain rate are obtained from Taylor [20]. Isothermal Flow Simulations With the intent of further confirming the predictive capabilities of the LES model for incompressible and low-Mach-number isothermal flows, simulations at Re 4 15,100, 22,100, and 37,100, corresponding to the experiments in Ref. [19], are performed. In addition, these simulations are intended to provide a better understanding of rearward-facing step flow in general, and since the topology of this flow is mainly preserved under combustion, a discussion of its flow field can lead to an improved understanding of the reacting flow. The flow can be separated into the mixing layer region, comprising of the curved shear-layer flow, from the initial boundary layer at separation to the reattachment point, and the relaxation region, from the reattachment point to full recovery of the turbulent boundary layer (Fig. 1b). The rearward-facing step flow develops similarly to a free shear layer with zero velocity on one side; before curvature effects become large, the growth rate and velocity profiles are

A FLAME-WRINKLING LES COMBUSTION MODEL

similar to the half-jet (cf. Ref. [21]). However, unlike the half-jet, the upper velocity varies in the streamwise direction, and more importantly, the velocity in the recirculation region is not zero. As the layer continues to develop, curvature effects become large, and the mixing layer analogy no longer applies. The layer curves down toward the lower wall and impinges on the wall at the reattachment point. The size of the recirculation bubble depends to a large extent on the rate of growth of the reattaching shear layer into the recirculation zone, see profiles in Fig. 2. As in mixing layers, the topological structure of the reattaching shear layer is determined by the development of large-scale coherent structures formed by the Kelvin–Helmholtz instability in the shear layer (Fig. 1b). The resulting structures are predominantly two dimensional and grow by fluid entrainment and coalescence. The growth of these structures affects the recirculation zone as well as the rate of spread of the upper boundary of the shear layer into the free stream. Moreover, the turbulence in the layer results from the passing of large-scale structures containing smaller-scale, three-dimensional structures. In Fig. 2, averaged profiles of the streamwise velocity component and its rms fluctuations are shown for the Re 4 22,100 case. Experimental and LES results agree favorably for both the streamwise velocity component and its rms fluctuations. Specifically, it should be noticed that LES satisfactorally captures the high maximum reverse velocity of 0.32u0 (0.33u0) (the experimental value given in parentheses), the length of the recirculation region xR/ h 4 7.1 (7.0), and the normalized growth rate of the mixing layer, defined as d 4 dU¯/(hdU¯1/dx2), where dU¯1 is the velocity difference across the layer, which is approximately constant between x1/h 4 1 and 3 with a value of 0.27 (0.28).

Reacting Flow Simulations To validate the combustion model and to advance our understanding of premixed turbulent reacting flow, simulations at Re 4 22,100, corresponding to the experiments [18,19], are carried out. Some aspects of the flow topology are illustrated in Fig. 3. LES is successful in capturing the global flow features, in particular, the shape of the initial shear layer and the downstream evolution of large-scale structures. In the reacting case, the incoming fluid contains cold premixed reactants, which mix with hot combustion products in the initial shear layer behind the step prior to burning. The large strain rates in this region delay ignition of the flame, allowing the development of a Kelvin–Helmholtz instability. Exothermicity occurs mainly in the large-scale structures that form early in the shear layer as they entrain cold reactants and hot products. The resulting

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volumetric expansion within the structures contributes to the growth of the shear layer d 4 0.30 (0.29) (the experimental value is given in parentheses), which is well reproduced by LES. Because there is negligible reduction in growth rate from the isothermal to the reacting case, the effect of the density and viscosity differences across the shear layer (which are expected to reduce the growth rate) appears to be counterbalanced by the volumetric expansion. It is found that if the laminar flame speed is assumed independent of strain rate, the heat release in the shear layer inhibits the growth of the instability, resulting in a near smooth flame. From this and the experimental observation that acoustic resonance is not a dominant feature of this case [18], it may be concluded that the magnitude of the shearlayer instability in the reacting case is strongly affected by the sensitivity of lean propane flames to strain rate and that a richer flame would exhibit a smaller instability. Figures 3c and 3d show visualizations of the vorticity magnitude |x ¯ | and the effective reaction rate ˜ Topologically, x x ˙ 4 quSuN|¹b|. ¯ seems to be characterized by spanwise Kelvin–Helmholtz vortices induced by the shear-layer instability, with longitudinal vortices stretched between. The Kelvin–Helmholtz vortices are found to undergo helical pairing and experience intense vortex stretching as they are convected downstream. The longitudinal vortices develop in the “braided” region between adjacent spanwise structures and have a ropelike topology. Far downstream, vortex stretching effects break down the spanwise structures, while the longitudinal structures maintain their coherence. In the isothermal case (Fig. 1b), such features dominate the flow, whereas in the reacting case, they combine with specific combustion characteristics: volumetric expansion and density gradient effects resulting in a less structured and more diffuse vorticity field, reducing the straining of the reaction interface and the mass entrainment rate. The effective reaction rate is mainly confined to topological arrangements of sheetlike structures that are folded into the cores of the spanwise vortices where rapid burning takes place. The longitudinal vortices appear mainly to wrinkle the reaction sheet causing the surface to develop regions of high curvature. Figures 2a and 2b show averaged profiles of the streamwise velocity component and its rms fluctuations. For the isothermal case, the LES profiles are in good agreement with the experimental data, while for the reacting case, the agreement is reasonable. The length of the recirculation zone is found to be shorter in the reacting case, and the maximum reverse velocity is higher due to gas expansion. The LES models are successful in reproducing these features, the maximum reverse velocity being 0.41u0, 0.42u0, and 0.45u0 (0.44u0), and the length of the

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(a)

(b)

(c)

(d)

Fig. 2. Time-averaged profiles of the (a) streamwise velocity component, (b) streamwise rms-velocity fluctuations, (c) temperature, and (d) CO2 percentage volume fraction. Legend: (`) experimental and (1) LES results for the isothermal Re 4 22,100 case [19], (C) experimental data for the reacting Re 4 22,100 case [18,19], (– –) LES results using the algebraic model, (-•-) LES results using the one-equation model, and (—) LES results using the two-equation model.

A FLAME-WRINKLING LES COMBUSTION MODEL

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Fig. 3. Perspective view of the computational domain showing (a) a Schlieren photograph obtained from Ref. [22] having an exposure time of 49 ls, (b) the temperature averaged over 50 ls, (c) the instantaneous vorticity magnitude |x|, ¯ and (d) the effective reaction rate x. ˙ For these results, the two-equation model, that is, equation 3, in conjunction with equations 5 and 12, is used.

recirculation region is xR/h 4 4.3, 4.4, and 4.6 (4.5), for the algebraic, one-equation, and two-equation models, respectively. The growth rate of the mixing layer is also correctly reproduced by all three models. From Fig. 2b, it can be observed that the turbulence intensity is well captured by the LES models both in the isothermal and reacting cases. For both flows, regions of high turbulence occur in the shear layer and broaden with downstream distance from the step. Near the reattachment point, the intensities decrease and rms profiles begin to resemble those of fully developed turbulent channel flow. The averaged temperature profiles, Fig. 2c, show good agreement between LES and the experimental data and emphasize some particular features of the reaction region. The steep gradient at x1/h 4 0.01 corresponds to the early shear layer, which gradually rolls up in a sequence of large-scale structures, thus broadening the temperature profiles. In Fig. 2d, averaged CO2 percentage volume fraction profiles (obtained from b¯ assuming complete combustion) are presented, again showing good agreement between LES and measurement data.

Distinct differences between the algebraic, oneequation, and two-equation versions of the model are apparent from the profiles in Fig. 2. These differences are consistent with the observation that the flame instability is best reproduced by the two-equation model (cf. Fig. 3); the one-equation model and, in particular, the algebraic model result in a weaker instability. Currently, this is believed to relate to a deviation from local equilibrium and the lack of time-history effects in the simpler models. It is clear that strain effects are particularly important for lean propane flames and that these are not handled appropriately by the equilibrium models. Conclusions A new LES flame-wrinkling combustion model, derived by the application of conditional filtering to the turbulent flame front, is proposed and validated against experimental data from a reacting shear layer formed at a rearward-facing step. The approach may be viewed as a hierarchy of models with increasing

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complexity and physical justification, ranging from a quasi-algebraic model in which a transport equation for the regress variable b˜ is solved using algebraic expressions for the flame-wrinkling N and the laminar flame speed Su, to a two-equation model, in which N and Su are also obtained from modeled transport equations. To close these equations, models are developed for the conditionally filtered unburned gas velocity, the unburned–burned gas slip velocity, and the surface-filtered velocity of the flame. In addition, the subgrid turbulent generation and removal rates are represented by algebraic expressions derived from a set of spectral evolution equations. Simulation of the validation case with the proposed models yields good overall agreement, with the two-equation model giving the best results. In particular, the stretch effects on the lean premixed flame are found to be important in that high strain causes local extinction during roll-up in the initial shear layer, thus delaying heat release and volume expansion, resulting in the characteristic largescale structures. To handle these effects properly, the full two-equation model is required, in which the Su equation successfully handles both time-history and strain effects. Acknowledgments This work is supported by the EPSRC under Grant Number K20910.

REFERENCES 1. Maly, R. R., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 111–124. 2. Correa, S. M., Hu, I. Z., and Tolpadi, A. K., J. Energy Res. Technol. Trans. ASME 118(3):201–208 (1996). 3. Strahle, W. C., in Nineteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1983, pp. 337–347. 4. Grinstein, F. F. and Kailasanath, K. K., Combust. Flame 100:2–10 (1995). 5. Menon, S. and Kerstein, A. R., in Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 443–450.

6. Mo¨ller, S. I., Lundgren, E., and Fureby, C., in TwentySixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 241–248. 7. Piana, J., Ducros, F., and Veynante, D., Turbulent Shear Flows 11:21–13 (1997). 8. Libby, P. A. and Williams, F. A., in Turbulent Reacting Flows, Topics in Applied Physics, vol. 44, Lecture Notes in Physics (P. A. Libby and F. A. Williams, eds.), Springer-Verlag, New York, 1980. 9. Fureby, C. and Lo¨fsto¨m, C., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994. 10. Fureby, C., Tabor, G., Weller, H. G., and Gosman, A. D., Phys. Fluids 9(5):1416–1429 (1997). 11. Weller, H. G., Thermo-Fluids Section report TF 9307, Imperial College of Science, Technology and Medicine, March 1993. 12. Weller, H. G., Uslu, S., Gosman, A. D., Maly, R. R., Herweg, R., and Heel, B., in International Symposium COMODIA 94, The Japan Society of Mechanical Engineers, 1994, pp. 163–169. 13. Heel, B., “Dreidimensionale Simulation der Stro¨mung und Verbrennung im Zylinder eines Otto-Forschungsmotors,” Ph.D. thesis, Universita¨t Karlsruhe, 1997. 14. Weller, H. G., Marooney, C. J., and Gosman, A. D., in Twenty-Third Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1990, pp. 629–636. ¨ . L., in Twenty-Third Symposium (Interna15. Gu¨lder, O tional) on Combustion, The Combustion Institute, Pittsburgh, 1990, pp. 743–750. 16. Poinsot, T., Veynante, D., and Candel, S., J. Fluid Mech. 228:561–606 (1991). 17. Jasak, H., “Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows,” Ph.D. thesis, Imperial College of Science, Technology and Medicine, 1996. 18. Ganji, A. R. and Sawyer, R. F., AIAA. J. 18(7):817–824 (1980). 19. Pitz, R. W. and Daily, J. W., AIAA. J. 21(11):1565–1570 (1983). 20. Taylor, S. C., “Burning Velocity and the Influence of Flame Stretch”, Ph.D. thesis, University of Leeds, 1991. 21. Eaton, J. K. and Johnson, J. P., Technical report MD39, Stanford University, 1980. 22. Pitz, R. W. and Daily, J. W., NASA Contractor Report 165427, 1981.

COMMENTS Jerzy Chomiak, Chalmers University of Technology, Sweden. In your simulation, you used a complex subgrid model. Comment on the sensitivity of the results to selection of the particular modeling constants. Author’s Reply. The subgrid model consists of two parts: the first relates to the unresolved transport terms and is

closed by a one-equation eddy-viscosity model, using a separate transport equation for the subgrid kinetic energy [1]. The model coefficients follow from a |k|15/3 inertial subrange behavior, and thus, no free parameters appear in this model. The subgrid flame-wrinkling combustion model is based on the solution of three modeled transport equations: the b-equation requires Xi and Su, the Xi-transport

A FLAME-WRINKLING LES COMBUSTION MODEL equation is closed using the turbulent flame-speed correlation of Gulder (Ref. [15] in the paper), validated using the spectral approach of Weller (Ref. [14] in the paper), and contains no additional free parameters, while the Suequation is modeled using asymptotic relaxation based on a chemical and a strain-rate time scale, and thus, no free parameters appear in this model.

REFERENCE 1. Yoshizawa, A., Phys. Fluids A 29:2152 (1986). ● Norbert Peters, Universita¨t Aachen, Germany. This is certainly a very interesting piece of work. Is there a physical basis for the transport equation for the laminar burning velocity? It seems to me that it would need to contain an analysis of the reactive-diffusive structure of the premixed flame.

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Author’s Reply. In this work, the laminar-flamelet approach is used in which the flame surface is considered to be an infinitesimally thin sheet propagating normal to itself at the local laminar flame speed Su, which is considered to be a function of the thermodynamic state of the unburned gas, strain rate, and curvature. To represent transport and history effects, a transport equation for Su is hypothesized in which the appropriate substantive derivative is that which transports all properties of the flame surface, as derived for the equation for Xi, and the sources modeled using asymptotic relaxation based on a chemical and a strainrate time scale. This model requires the unstrained Su and strain response as input, both of which are the result of an analysis of the reactive-diffusive structure of the premixed flame (e.g., via 1-D full kinetics laminar flame calculations). It may also be possible to derive a similar transport equation for the laminar flame speed from an analysis of the reactive-diffusive structure of the premixed flame, but considering the model assumes the flame surface is infinitesimally thin, it is not clear to the authors how to proceed.