A computational study of combustion instabilities due to vortex shedding

Proceedings of the Combustion Institute, Volume 28, 2000/pp. 783–791

A COMPUTATIONAL STUDY OF COMBUSTION INSTABILITIES DUE TO VORTEX SHEDDING C. FUREBY Department of Weapons and Protection Division of Warheads and Propulsion FOA Defence Research Establishment S-172 90 Stockholm, Sweden

The influence of unsteady vortex shedding on combustion instabilities is examined using simulation databases for two configurations: a dump combustor and a model afterburner equipped with a bluff-body flame holder for different operating conditions. These databases have been generated using a recently developed flame-wrinkling large eddy simulation (LES) model, described elsewhere. As a first necessary step, a detailed comparison is made between predicted and experimentally obtained statistics to quantify the accuracy of the LES and determine whether LES reproduces the observed statistical trends in the experiments. The objective of the present study is to examine these databases using visualization techniques in order to investigate the mechanisms responsible for combustion instabilities in relation to unsteady vortex phenomena. Combustion instabilities are found, experimentally as well as computationally, in both configurations and under different operating conditions. Based on the LES databases, the origin of these instabilities can be identified and attributed to key events such as vortex shedding, excitation of pressure fluctuations due to exothermicity, extinction by strain, and operating conditions. The unsteady behavior of the dump combustor is dominated by extinction due to the high strain rate, while the dynamics of the model afterburner was associated mainly with excitation of pressure fluctuations due to exothermicity and operating conditions.

Introduction The development of future combustors, including lean premixed prevaporized (LPP) combustors and ramjets, in which the flame is stabilized in the wake of a flame holder or a rearward facing step, requires better understanding of combustion instabilities, unsteady vortex dynamics, ignition, flashback, and flame-holding capacity. The operation of such devices is often impaired by potentially harmful combustion instabilities, which at the lean limit may cause blow off, and at the rich limit may lead to flashback. In dump combustors and combustors with bluff-body flame holders, vortices are formed in the shear layer, separating regions of high-speed flow from regions of low speed flow due to Kelvin–Helmholtz instabilities. In general, the high-speed flow is composed of an unburned mixture of fuel and air while the low-speed stream is composed of hot combustion products, forming the flame holder recirculation region behind the dump plane or the bluffbody. Turbulent mixing after the trailing edges of a bluff-body flame holder or behind the step in a dump combustor together with the effects of recirculation are known to be important to the stability of the flame and have been successfully examined in different experimental facilities (e.g., Refs. [1–3]). Computational studies are often restricted by the 783

limitations of the Reynolds average simulation (RAS) models [4] and the turbulent combustion models used to represent the mean flame and its effects on the flowfield. The concept of large eddy simulation (LES) [5] thus provides a more natural approach to unsteady flows with ranges of parameters similar to those of practical or engineering-like systems [6]. The present study is concerned with the application of LES to examine the key steps of combustioninduced instabilities and the associated unsteady vortex dynamics. To this end, two laboratory devices, a dump combustor [7] and a model afterburner with a triangular-shaped bluff-body flame-holder [8], have been selected as well-characterized and wellbehaved representatives of real combustors. For both facilities, extensive experimental [7,8] and LES [9,10], databases for non-reacting and reacting conditions exists. The LES databases are obtained using a recently developed flame-wrinkling LES combustion model [9,10], carefully validated against firstand second-order statistical moments of velocity, temperature, and species concentrations using different experimental sources. The main objective of the present study is to use the LES databases to obtain a more complete understanding of the physics behind combustion-induced instabilities due to vortex shedding. It should be pointed out that the discussion of the results draws on a deep analysis of the

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extensive material in the LES databases, and selected figures are used only to illustrate some general features of the combustion dynamics. A further objective is to establish how well the LES flamewrinkling model can represent different modes of operation found in the laboratory experiments. In this work, we discuss flowfield parameters such as vorticity, density, and rate-of-strain and how they are related to key phenomena such as ignition, flashback, and flame stabilization.

The Computational Approach With LES, we attempt to resolve most of the entrainment-dominating large-scale coherent structures (CSs) by choosing the cutoff wavelength within the inertial subrange. The challenge is to correctly mimic the fluid dynamics near the cutoff and to ensure proper interactions between resolved and unresolved eddy scales. The LES equations describe the evolution of the large-scale features of the flow and are derived by applying a low-pass filter to the balance equations of mass, momentum, and energy, given appropriate constitutive equations. The effects of the unresolved eddies result in additional unknown subgrid terms in the LES equations that must be separately modeled. Subgrid models for non-reacting LES have been developed in the past [11], but few extensions to reacting flows have been made [9,10,12–14], since additional closure problems arise from the combustion-related terms which are difficult to model. However, the unresolved transport terms are not unique to reacting flows and can be modeled by conventional subgrid models. A recent alternative is monotone integrated LES (MILES) [15], in which the reacting Navier–Stokes equations are solved by high-resolution locally monotonic algorithms with built-in low-pass filtering and subgrid turbulence models. In this study, we use a flamewrinkling LES combustion model [9,10] to examine the reacting flow in the previously described combustor configurations. In the flame-wrinkling LES combustion model, the reaction is measured with a reaction coordinate b satisfying
t(qb) Ⳮ div (qvb) ⳱ div (D grad b) Ⳮ m ˆ where v is the velocity; q, density; D, effective diffusivity, and m ˆ , mass supply term. If the diffusion– reaction terms are combined and represented by qSu|grad b| where Su is the laminar flame speed, a Hamilton– Jacobi equation is obtained. If the governing equations, that is, the balance equations of mass, momentum, and energy, and the Hamilton–Jacobi equation for b are low-pass filtered using the kernel

G, the LES equations result. The unresolved transport terms are not unique to reacting LES, and ordinary subgrid models, such as the one-equation eddy–viscosity model [16], can therefore be used to close the LES equations and to model the effects of the subgrid eddy scales on the resolved flow. The definition of the Favre´-filtered reaction coordinate b˜ provides a suitable measure of the large-scale geometry of the flame, but to accurately simulate flame propagation we must also consider the subgrid flame wrinkling. It can be shown [9] that the filtered source term in the Hamilton–Jacobi equation can be decomposed as ¯ qSu|grad b| ⳱ qSu R where  •  denotes surface filtering, and ¯ ⳱ |grad b| R is the amount of flame surface within the support of the filter kernel G. By introducing the flame-wrinkling density ¯ ¯ N ⳱ R/|grad b| where ¯ |grad b| is the area of the resolved flame surface, we may take the decomposition one step further, so that ¯ qSu|grad b| ⳱ qSuN|grad b| From the definition of N, it is evident that it represents the amount of surface per unit filtered flame surface. Based on the true balance equation for N, a modeled balance equation has been proposed [9]. By decomposition into total and surface strain rates rT and rI, respectively, and by introducing the subgrid generation and removal rates G and R, this model becomes
t(N) Ⳮ vI • N ⳱ GN ⳮ R(N ⳮ 1) Ⳮ N(rI ⳮ rT) where vI is modeled as ˜ ˜ slip Ⳮ Sun/N vI ⳱ v˜ Ⳮ (1 ⳮ b)v and n is the unit normal vector of b. Following [9– 10], v˜ is the LES velocity, v˜slip is the velocity difference over the flame due to the density ratio and the subgrid turbulence, and Sun/N represents the effects of differential propagation and cusp formation. The turbulence–flame interaction model uses the flame speed model of Gu¨lder [17], resulting in expressions for G and R [9]. To account for strain and curvature, a transport equation for the flame speed Su is introduced [9], under the assumption that Su is advected with the flame velocity and affected only by the resolved strain rate and chemical time scale. The LES model finally consists of low-pass filtered equations of mass, momentum, and energy; the

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Hamilton–Jacobi equation for b; and model equations for N and Su. To close the equations and to model the subgrid turbulence, a one-equation eddy viscosity model [16], is used. The laminar flame speed and the associated strain response data are derived from databases to correspond to the freestream flow conditions. The governing equations are discretized using an unstructured finite volume method. Second-order schemes are used in space and time, central differencing for velocity, a bounded normalized variable diagram (NVD) scheme for scalars [18], and a threepoint backward differencing in time. To decouple the pressure–velocity system, a Poisson equation, obtained from the discretized continuity and momentum equations, is constructed for the pressure. The set of scalar equations are solved sequentially with iteration over the explicit coupling terms to obtain rapid convergence. The segregated approach results in a Courant number restriction; a maximum Courant number of 0.5 gives satisfactory numerical stability and temporal accuracy, but a value of 0.2 is preferable for temporal accuracy.

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Dump Combustors

Fig. 1. Typical profiles of (a) v˜1, ˜ ˜ (b) vrms 1 , (c) T, and (d) YCO2 for the dump combustor at x1/h ⳱ 5.0. ——, LES of non-reacting case; —, LES of reacting case at 170 ⳯ 80 ⳯ 40; - • -, LES of reacting case at 255 ⳯ 120 ⳯ 60; Ⳮ, experiments of non-reacting case; and 䡩 experiments of reacting case.

The dump combustor chosen has been used by Ganji and Sawyer [1], Keller et al. [2], and Pitz and Daily [7], and experimental data exist for various operating conditions. It consists of a rectilinear premixing section followed by a contraction to one half of its height h continued by a step expansion into the combustor, while in the simulations only the expansion and the combustor section were considered. In the LES, appropriate boundary conditions are given at the inlet (propane/air mixture with an equivalence ratio of ␾ ⳱ 0.57, p0 ⳱ 101 kPa, u0 ⳱ 13.3 m/s, and T0 ⳱ 293 K), outlet, top and bottom walls, while periodic conditions are enforced in the spanwise direction (see Ref. [9]. All simulations are initialized with quiescent conditions, and the unsteady flow evolves naturally. The effective computational domain size is 16 h ⳯ 2 h ⳯ 4 h in the streamwise, spanwise, and cross-stream directions, and the grids typically use between 170 ⳯ 80 ⳯ 40 and 255 ⳯ 120 ⳯ 60 cells, refined in the vicinity of the top and bottom walls and stretched in other parts of the domain. Non-reacting and reacting LES have been presented separately [9,19], and here we use these LES databases to examine the coupling between unsteady coherent vortical structures, exothermicity, and acoustics. Figure 1 presents typical profiles of the time-averaged streamwise velocity v˜1, its (rms) root mean square fluctuations v˜ rms 1 , the time-averaged temperature T˜ and the time-averaged CO2 mass fraction Y˜ CO2, for the non-reacting and reacting cases,

respectively. The Y˜ CO2 is determined from b˜ and the initial mass fractions corresponding to the equivalence ratio. The agreement between the predictions [9], and the experiments [7] is sufficiently good to warrant this study, although some differences can be observed in the v˜ rms 1 profile for the reacting case. The length of the recirculation region is shorter in the reacting case (l/h ⳱ 4.6) as compared to the nonreacting case (l/h ⳱ 7.2) and in good agreement with the experimental data of l/h ⳱ 4.5 and 7.3, respectively. Regions of high-velocity fluctuations are confined by the shear layer which widens with downstream distance from the step. In particular, the peak values of ˜vrms increase initially due to the 1 formation of large CSs, to finally stabilise around 20% of u0, being in good agreement with experiments [7]. Profiles of T˜ and Y˜ CO2  show good qualitative and quantitative agreement between simulations and experiments, although minor differences in the profile shape are observed for Y˜ CO2. Global quantities, such as the vorticity thickness, Strouhal number, entrainment, and shear layer growth rate, are all very well predicted by the LES model. Moreover, the actual growth of the shear layer is delayed in the reacting case as compared to the non-reacting case due to the temperature-dependent viscosity. Figure 2 shows schlieren images from Pitz and

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Fig. 2. Experimental and computational visualization of the dump combustor using (a) schlieren [7], (c) an isosurface of k2 to illustrate the vorticity, (b) semitranslucent isosurfaces of b˜ to emulate the schlieren image, and (d) contours of the effective reaction rate combined with iso-surfaces of the peak values. The computational results are selected from a typical cycle in an animation to match the experimental schlieren image.

˜ vorticity, Daily [7], semitranslucent iso-surfaces of b, and the effective reaction rate. Following Jeong and Hussain [20], the vorticity is represented as an iso surface of k2 (for values smaller than zero), where k2 is the intermediate eigenvalue of the tensor L2 ⳱ D2 Ⳮ W2, where D is the rate-of-strain tensor and W the rate-of-rotation tensor. This interpretation of a vortex is based on the fact that L2 determines the existence of a local pressure minimum due to vortical motion. By comparing schlieren images with ˜ or the density q, we see that the animations of b, LES model is successful in capturing the flow, in particular the shape of the shear layer and the development of large CS. The spanwise vortices that shed of the step develop as a result of the rollup of the boundary layer. Due to concentration of vorticity, longitudinal vortices develop together with undulations of a newly shed spanwise vortex in areas of

high-strain between subsequent vortices of the same orientation. As spanwise and longitudinal vortices are advected downstream, under the influence of vortex stretching they reconnect and merge in a specific manner [19], forming a mixed pattern of X and big K vortices that impinge on the lower wall and are either carried away downstream and transformed into arches or become trapped in the recirculation bubble. As result of the vortical flow, the growth of CSs with downstream distance from the step affects the recirculation region and the rate of spread of the upper boundary of the shear layer into the freestream. Volumetric expansion due to exothermicity, baroclinic torque effects, and temperature-dependent viscosity combine to form thicker vortical structures, as compared to the non-reacting flow, thus reducing the strain of the flame and the mass entrainment rate. The effective reaction rate

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¯ w ¯ ⳱ quSuN |grad b| is primarily confined to topological arrangements of sheetlike structures, which fold into the cores of the vortices where rapid burning takes place. The longitudinal vortices mainly wrinkle the reaction sheet, and hence this surface develops regions of high curvature. As the flame is advected downstream, it propagates normal to itself at the flame speed SuN, causing negatively curved wrinkles to contract and positively curved wrinkles to expand. This also increases the possibility of pockets of reactants breaking away from the rest of the reactants. As spanwise vortices shed of the step rollup, cold premixed reactants and hot combustion products become entrained and mix macroscopically. Reaction is, however, suppressed by the high strain in the shear layer after which the reaction continues, causing further volumetric expansion and intense burning, forming the characteristic large-scale pattern of burning CSs.

Combustors with Bluff Body Flameholders The bluff-body flame holder case consists of a rectilinear duct, with rectangular cross-section, that is divided into an inlet and a combustor section equipped with a two-dimensional triangular-shaped flame holder [8]. Results will be discussed for a nonreacting case characterized by u0 ⳱ 17 m/s and T0 ⳱ 293 K (case I) and two reacting cases using a lean propane/air mixture (␾ ⳱ 0.65 at p0 ⳱ 101 kPa), characterized by u0 ⳱ 17 m/s and T0 ⳱ 288 K (case II) and u0 ⳱ 34 m/s and T0 ⳱ 600 K (case III), respectively. In the simulations, appropriate boundary conditions are specified at the inlet, outlet, upper and lower walls, while periodic conditions are enforced in the spanwise direction. All three simulations are initialized with quiescent conditions, allowing the unsteady flow to evolve by itself. The size of the domain is 20 h ⳯ 3 h ⳯ 3 h (where h is the bluff-body height) in the streamwise, spanwise, and cross-stream directions, and different grids, employing between 200 ⳯ 30 ⳯ 80 and 300 ⳯ 45 ⳯ 120 cells, have been used. The grids are refined in the vicinity the walls and around the flame holder and stretched elsewhere. Further details from these simulations are presented elsewhere [10]. Figure 3 shows a typical comparison between predicted and experimental data for the non-reacting case I, as well as for the reacting cases II and III. The general agreement for v˜1 and v˜ rms is good, al1 though it is better for the non-reacting case than for the reacting cases. Comparing profiles of v˜1, with and without combustion, given the same airflow, shows that the reacting cases have a longer and wider recirculation region and, hence, a more gradual dissipation of momentum in the wake region as compared to the non-reacting cases. The simulations

Fig. 3. Typical profiles of (a) v˜1, ˜ ˜ (b) vrms 1 , (c) T, and (d) YCO2 for the bluff-body flame holder case at x1/h ⳱ 11.5. - • - • , LES of case I; —, LES of case II; —, LES of case III; Ⳮ, experiments of case I, 䡩, experiments of case II, and ▫, experiments of case III.

successfully capture the differences in v˜1 between the non-reacting and reacting flows, and more importantly, the differences between cases II and III. Case III results in the widest and longest recirculation region; the reason for this is mainly due to the higher flame speed and different density ratios, resulting from the preheated inflow, but is also due to different levels of flame-generated turbulence. Far downstream, the velocity profiles parallel those of fully developed turbulent channel flow but having a bulk velocity about twice the inlet velocity due to the volumetric expansion caused by exothermicity. The streamwise rms-velocity fluctuations vrms are prop1 erly predicted, and the differences between case II and III are here very distinct. The predicted profiles of T˜ are in reasonable agreement with the experimental data from gas analysis [8] and coherent antiRaman scattering (CARS) [21], and hence the predictions are within the overall uncertainty levels of the measurements, determined by the difference between measurement data from gas analysis and CARS. The difference between cases II and III is evident also from the T˜ profiles, and further emphasized in the probability density functions (PDFs) of T˜, where case III shows bimodal PDFs in the shear layers [10]. Concerning profiles of Y˜ CO2, the agreement with the measured data is reasonable for both cases, but with some differences far downstream. Global quantities, such as the length of

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Fig. 4. Computational visualisations of the bluff-body flame holder cases II (a) and III (b), by means of an isosurface ˜ and contours of the Rayleigh parameter R in copperplate of k2 to illustrate the vorticity, semitranslucent isosurfaces of b, style ⳱ cov[p¯, Q] at the centerplane. Gray contour lines denote regions with R in copperplate style  0, and black contour lines denote regions with R in copperplate style  0, representative of driving.

the recirculation region and the Strouhal (St) number, are very well predicted (see Ref. [10]), for the non-reacting as well as for the two reacting cases. Figure 4a and b shows iso-surfaces of b˜ and k2 and contours of the covariance between the pressure and the heat release for cases II and III, respectively. In both cases, the flame is essentially confined to topological arrangements of sheetlike structure, emanating in the shear layers, and stabilized through recirculation of hot combustion products in the near wake. The general flow features include spanwise vortices being shed of the edges of the prism, either simultaneously as in case II or alternating as in case III, and rollup while advected downstream, under the influence of vortex stretching, allowing reactants and combustion products to mix macroscopically before burning. For case II, LES predicts a varicoselike behavior, with symmetric shedding of spanwise vortices, and characterized by longitudinal oscillations. For case III, a sinuous-like behavior is obtained that renders large-scale antisymmetric shedding with every other spanwise vortex being shed from the upper (or lower) edge of the prism. Depending on if the spanwise vortices are shed simultaneously (as in case II) or alternating (as in case III), the resulting vortex interaction processes will be different, before being further modified by the exothermicity. For case II, the simultaneous shedding and rollup results in longitudinal vortices stretched between sequential spanwise vortices on either side of the centerline of the wake. For case III, the alternate

shedding and rollup results in vortex dynamics similar to that of the non-reacting case with longitudinal vortices stretched between successive spanwise vortices of alternate sign. In both cases, however, following the first rollup and the subsequent growth of longitudinal vortices, combustion-related effects, such as volumetric expansion, baroclinic torque effects, and increased molecular viscosity (due to exothermicity), combine to modify the vorticity, resulting in the development of multiple small-scale vortices with reduced vorticity magnitude in the core region, as apparent in the k2 distributions of Fig. 4. Following Putnam [22], it can be argued that the dynamics observed are due to the combined effects of vortex shedding and excitation of acoustic oscillations due to exothermicity. According to Rayleigh [23], this occurs when a proper phase relationship between the (periodic) heat release and pressure oscillations exists. To examine this we compare, in Fig. 4a and b, the Rayleigh parameter R in copperplate style ⳱ cov[p¯, Q], where p¯ is the pressure and Q the heat release, between cases II and III. For case II, driving occurs frequently in the shear layers between the spanwise vortical structures and in regions with positive curvature, while for case III, driving occurs less frequently in selected regions with positive curvature. In addition to this, other mechanisms, such as blow off, may be relevant to consider. By varying u0 and ␾, with reference to case II, we find that the varicose mode dominates at low-flow velocities, while for higher-flow velocities, near the blow off

COMBUSTION INSTABILITIES DUE TO VORTEX SHEDDING

limit, the sinuous mode is the preferred configuration of the flame. However, for the asymmetric flame, a symmetric region is observed near the prism, where the recirculation region exists.

Discussion and Concluding Remarks In this study, LES databases of two different laboratory combustors have been examined in order to gain additional insight into combustion instabilities and unsteady vortex dynamics. The simulations are performed with a flame-wrinkling LES combustion model [9,10], and first a quantitative comparison is made with experimental data from both combustors in order to verify the accuracy of the LES model. These comparisons give satisfactory agreement with the experimental data for first- and second-order statistical moments of velocity, temperature, and species concentrations, as well as for probability density functions of the temperature [10]. A qualitative comparison also shows that the LES model is capable of separating the statistics from two operating conditions being tested on the model afterburner. The LES suggests that the dynamics of these cases are different, which is confirmed by high-speed video photography [8]. Hence, the quality of the simulations is sufficient to use the databases for more indepth studies of basic physical phenomena. This study does not provide combustor design criteria, per se, but insight into the underlying physics which can guide the designer in the development of combustors and passive control methods. The shed vortices behind the dump plane in a dump combustor or behind a bluff-body flame holder grow and merge as they are convected downstream under the influence of vortex stretching, exothermicity, and volumetric expansion. Due to merger and entrainment, the shear layer widens and may interact with other shear layers, being either in or out of phase, depending on the confinement and the operating conditions. Non-reacting simulations of the dump combustor and the bluff-body flame holder combustor supports the notion that Kelvin– Helmholtz instabilities are responsible for rollup of vortices in the shear layers, having a frequency that scales with the characteristic length scale and the flow speed. The predicted pressure spectrum of the dump combustor displays several coexisting instabilities at frequencies ranging from 20 to 400 Hz. For the bluff-body flame holder the situation is more complex; case II shows longitudinal pressure oscillations at 100 Hz, causing the shear layers to rollup in a symmetric manner, resulting in periodic heat release that feeds energy back into the pressure. In case III, the time history of the cross-stream velocity component indicates vortex shedding, due to vortices being shed from alternate sides of the prism at about 120 Hz. For this case, the pressure spectrum

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of shows peaks at 120 and 950 Hz, suggesting that this mode is a superposition of a low-frequency longitudinal and a high-frequency transverse mode. The development of large-scale coherent structures enhances macroscopic mixing (eddy scales larger than the typical grid size D) in the shear layer(s), although it hinders microscopic mixing (eddy scales smaller than D), necessary to initiate combustion. A more detailed analysis of the databases shows that in the presence of large coherent structures transition to microscopic mixing is initiated in the braid region of the spanwise vortices where the high strain rates between the two streams prevail. However, production of fine-scale turbulence appears restricted to the interface between the two streams. Moreover, the turbulent production is enhanced during the rollup of the vortices and by the merging of vortices. In combustors with flame holders, multiple shear-layer interaction results in an increase in the flame wrinkling N, which in turn, increases the effective reaction rate, which is followed by rapid and local exothermicity, as is the case for the bluff-body flame holder configuration II. Flapping of the flame branches may also cause the flame to interact with the confinement, as is the case for the bluff-body flame holder configuration III, which also leads to intermittent and local exothermicity— often harmful for the combustor wall. Acknowledgment The author wishes to thank H. G. Weller for many fruitful discussions and for developing the multipurpose CFD code FOAM used in the present study.

REFERENCES 1. Ganji, A. R., and Sawyer, R. F., AIAA J. 18:817–824 (1980). 2. Keller, J. O., Vaneveld, L., Korschelt, D., Hubbard, G. L., Ghoniem, A. F., Daily, J. W., and Oppenheim, A. K., AIAA J. 20:254–262 (1982). 3. Smith, D. A., and Zukoski, E. E., “Combustion Instability Sustained by Unsteady Vortex Combustion,” AIAA paper 85-1248, 1985. 4. Speziale, C. G., “Modeling of Turbulent Transport Equations,” in Simulation and Modeling of Turbulent Flows, (T. B. Gatski, M. Y. Hussaini, and J. L. Lumley, eds.), Oxford University Press, New York, 1996. 5. Reynolds, W. C., Turbulence at the Crossroads, (J. L. Lumley ed.), Springer Verlag, 1992, p. 313. 6. Bray, K. N. C., Proc. Combust. Inst. 26:1–26 (1996). 7. Pitz, R. W., and Daily, J. W., AIAA J. 21:1565–1570 (1983). 8. Sjunneson, A., Olovsson, S., and Sjo¨blom, B., “Validation Rig—A Tool for Flame Studies,” Report No. 9370-308 VOLVO Aero AB, S-461 81, Trollha¨ttan, Sweden, 1991.

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9. Weller, H. G., Tabor, G., Gosman, A. D., and Fureby, C., Proc. Combust. Inst. 27:899–907 (1998). 10. Fureby, C., Combust. Sci. Technol., in press (2000). 11. Lesieur, M., and Me´tais, O., Annu. Rev. Fluid Mech. 28:45–82 (1996). 12. Grinstein, F. F., and Kailasanath, K. K., Combust. Flame 100:2–10 (1994). 13. Menon, S., and Kerstein, A. R., Proc. Combust. Inst. 24:443–450 (1992). 14. Piana, J., Ducros, F., and Veynante, D., Turbulent Shear Flows 11, Grenoble, France, Sept. 8–10, 1997, p. 21–13. 15. Boris, J. P., Whither Turbulence? Turbulence at the Crossroads, (J. L. Lumly, ed.), Springer Verlag, Berlin, 1990, pp. 344–265.

16. Schumann, U., J. Comput. Phys. 18:376–404 (1975). ¨ . L., Proc. Combust. Inst. 23:743–750 (1990). 17. Gu¨lder, O 18. Jasak, H., Weller, H. G., and Gosman, A. D., Int. J. Numer. Methods Fluids 31:431–442 (1999). 19. Fureby, C., AIAA J. 37:1401–1410 (1999). 20. Jeong, J., and Hussain, F., J. Fluid Mech. 285:69–94 (1995). 21. Sjunnesson, A., Henriksson, P., and Lo¨fstro¨m, C., “CARS Measurements and Visualization of Reacting Flows in a Bluff Body Stabilized Flame,” AIAA paper 92-3650, 1992. 22. Putnam, A. A., Combustion Driven Oscillations in Industry, Elsevier 1971. 23. Rayleigh, J. W. S., Theory of Sound, Vol. II, Dover, New York, 1945.

COMMENTS Thierry Baritaud, IFP, France. Did you use the law of the wall for turbulence and combustion, or just a refined grid? Can you comment on the influence of the used procedure? Author’s Reply. In all the simulations reported, the grid was refined toward the combustor walls (including the prism wall) to obtain y Ⳮ ⳱ 895 ⳮ 10. The OEEVM subgrid model is then integrated all the way to the wall. For non-reacting cases, this approach has been discussed and examined elsewhere [1], but for reacting flows, no careful examination of this approach has been made. Depending on the character of the wall in the laboratory combustor, no suitable information is readily available for comparison. This is an important issue but, as yet, there is a general lack of understanding of how to handle wall boundary conditions well in non-reacting LES, an issue even more complicated in reacting flows where reactions may be influenced at the wall by, for example, catalysis.

Author’s Reply. In the flame-wrinkling model, the laminar-flamelet approach is adopted in which the flame propagates normal to itself at the local laminar flame speed Su. Su is considered to be a function of the thermodynamic state of the reactants, rate-of-strain, and curvature. To represent transport and hereditary effects, an evolution equation for Su is hypothesized in which Su is assumed to be advected with the surface averaged interface velocity vI influenced by chemical and strain-rate timescales, as modeled using asymptotic relaxation. The model requires the unstrained flame speed SuO and the strain-response sect as input, both of which may be the result of an analysis of the reactive-diffusive structure of the premixed flame (e.g., via one-dimensional full kinetics laminar flame simulations). The model has been used to study several different flames [1,2] from which it is evident that this model is superior to its equilibrium versions. The numerical results per se are not influenced by this model. It may also be possible to derive a similar equation from the reactive-diffusive structure, but it is not clear how to proceed.

REFERENCE 1. Fureby, C., AIAA. J. 37:1401 (1999). REFERENCES ● Arnaud Trouve, George Washington University, USA. The flame initiation and stabilization model used in your study is based on a transport equation for the local flame speed. This model seems empirically based, and its domain of validity remains unknown. Could you please comment on the general quality of that model and on the sensitivity of your numerical results to this particular model.

1. Weller H., Tabor G., Gosman, A. D., and Fureby, C., Proc. Combust. Inst. 27:899–907 (1998). 2. Nwagwe, I., K., Weller, H. G., Tabor, G. R., Gosman, A. D., Lawes, M., Sheppard, C. G. W., and Wooley, R., Proc. Combust. Inst. 28:59–65 (2000). ●

COMBUSTION INSTABILITIES DUE TO VORTEX SHEDDING Jim O’Connor, Perkins Engines Company Ltd, UK. Can you expand on why you chose the “thin flame” approach for reacting flows? Author’s Reply. Previous experience with different LES combustion models [1,2], as well as theoretical considerations, suggests that for the cases under consideration, the FW-LES model is presently the best available candidate.

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REFERENCES 1. Moller, S.-I., Lundgren, E., and Fureby, C., Proc. Combust. Inst. 26:241–248 (1996). 2. Fureby, C., Grinstein, F. F., and Kailasanath, K., “Large Eddy Simulation of Premixed Turbulent Flow in a Rearward-Facing-Step Combustor,” AIAA paper 00-0863, 2000.