Sensitivity to the Presence of the Combustion Submodel for Large

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Sensitivity to the Presence of the Combustion Submodel for Large Eddy Simulation of Transient Premixed Flame
Vortex Interactions Valeria Di Sarli* and Almerinda Di Benedetto Istituto di Ricerche sulla Combustione—Consiglio Nazionale delle Ricerche (IRC
CNR), Via Diocleziano 328, 80124, Napoli, Italy ABSTRACT: In this paper, the sensitivity of large eddy simulation (LES) to the presence of the combustion submodel was investigated for transient interactions between premixed flame fronts and toroidal vortex structures generated at the wake of a circular orifice. To this end, LES computations were run, with and without the combustion submodel, for two orifice diameters: 40 mm and 20 mm. Nonuniform unstructured grids with a cell characteristic length varying in the range of 0.5
1 mm were used. In going from the 40-mm orifice to the 20-mm orifice, both the size and velocity of the vortex increase, leading to a different regime of interaction with the flame: the vortex only wrinkles the flame front in the 40-mm case (wrinkled regime) and also disrupts the continuity of the front, giving rise to the formation of separate reaction zones (i.e., flame pockets that leave the main front), in the 20-mm case (breakthrough regime). It has been found that the impact of the combustion submodel on LES predictions is strongly dependent on the regime of interaction. Results for the 40-mm orifice are substantially the same, regardless of the presence of the combustion submodel. Conversely, at the wake of the 20-mm orifice, the intensity of the flame
vortex interaction is such that the combustion submodel is strictly needed to reproduce both the qualitative (evolution of the pockets formed and their interaction with the main front) and quantitative (flame speed) characteristics of the flame propagation correctly.

1. INTRODUCTION Turbulent premixed flames, both steady and unsteady, are involved in several engineering applications (gas turbine combustors, industrial burners, spark ignition engines, boilers, furnaces, etc.). Such flames are also found in gas and vapor cloud explosions that represent a major issue for chemical and process industries. In practical situations, when an explosion occurs, the flame propagating away from an ignition source encounters obstacles (vessels, pipes, tanks, walls, flow cross-section variations, instrumentation, etc.) along its path. The unsteady coupling of the moving flame and the flow field induced by the presence of local blockage produces vortex structures of different length and velocity scales ahead of the flame front. The flame
vortex interaction leads the initially laminar flame to burn through various turbulent combustion regimes, which are dependent on both the size and velocity of the vortices encountered, thus accelerating the flame and increasing the rate of pressure rise.1 In order to understand and control the hazards and risks associated with gas and vapor cloud explosions, predictive computational fluid dynamics (CFD) models are required.2 Over the past decade, the rapid increase in computing power has led to considerable progress in the development of CFD techniques based on large eddy simulation (LES) for modeling turbulent reacting flows. LES is replacing classical (and less computationally demanding) methods based on Reynolds averaging (i.e., RANS methods) for its ability to provide a better representation of turbulence and, thus, the resulting flame
turbulence interaction. LES grasps the inherently time-dependent nature of turbulent flows and, therefore, is particularly fit for simulating transient combustion phenomena such as flame propagation during the course of explosions. This is confirmed in several literature works.3
8 r 2011 American Chemical Society

LES directly resolves the large vortex structures in a flow field, from the length scale of the computational domain down to a cutoff length scale linked to the grid cell size, and only models the subgrid structures (that, however, exhibit a more universal behavior). The grid resolution represents the scale separation between the length scales of computed (resolved) turbulence and those of modeled (unresolved) turbulence. As the grid size decreases, the unresolved contribution decreases up to the point of reaching the asymptote of direct numerical simulation (DNS), with all turbulent scales computed and no need for subgrid turbulence models.9 LES of turbulent reacting flows also needs a combustion submodel.10,11 Indeed, LES does not resolve the flame on the computational grid (the premixed flame thickness is smaller than the grid size used). Consequently, the flame and its interaction with the subgrid vortices have to be modeled, whereas the effects of the large vortices on the flame propagation are explicitly resolved. Recently, we investigated the effect of the grid resolution on the impact of the combustion submodel for LES of transient premixed flame
vortex interactions in gas explosions.12 To this end, LES computations were run, with and without the combustion submodel, on three nonuniform grids with a cell characteristic length (Δ) varying in the ranges of 2
3 mm, 1
2 mm, and 0.5
1 mm. Numerical predictions were compared to the experimental data acquired by Long et al.13 during the interaction of a propagating flame front with a toroidal vortex generated at the Special Issue: Russo Issue Received: September 9, 2011 Accepted: December 1, 2011 Revised: November 28, 2011 Published: December 01, 2011 7704

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wake of a circular orifice. Experiments were conducted by changing the orifice diameter, which directly controls the size and velocity of the vortex, from 40 mm to 30 and 20 mm.13 LES calculations were performed for the 30-mm orifice. For this orifice size, results show that the solution obtained with the finer grid (having a resolution of the same order of magnitude as the laminar flame thickness) is independent of the presence of the combustion submodel. Indeed, the amount of detail explicitly resolved by LES is such that, even without using any combustion submodel, the predictions calculated with this grid correctly match the experimental data, in both quantitative (flame speed and flow velocity) and qualitative (shape and structure of the flame front) terms. In this paper, our previous investigation was extended to the cases of the 40- and 20-mm orifices. The sensitivity of LES to the presence of the combustion submodel was thus quantified for different vortex sizes and velocities (i.e., different combustion regimes). Nonuniform grids with Δ = 0.5
1 mm were used.

2. LARGE EDDY SIMULATION (LES) MODEL The LES model of unsteady premixed flame propagation used in this work has been described and validated previously.6,8,12 Briefly, the model equations were obtained by filtering the governing equations for unsteady compressible flows with premixed combustion, i.e., the reactive Navier
Stokes equations for conservation of mass, momentum, energy and chemical species, joined to the constitutive and state equations. The species balance equation was recast in the form of a transport equation for the reaction progress variable, c (c = 0 within fresh reactants and c = 1 within burned products).14 The transport equation for c reads as follows: ∂Fc þ ∇ 3 ðFucÞ ¼ ∇ 3 ðFD∇cÞ þ ω_ c ∂t

ð1Þ

Here, the two left-hand side terms respectively correspond to unsteady effects and convective fluxes, and the two right-hand side terms respectively correspond to molecular diffusion and reaction rate. To filter the governing equations, a low-pass box filter in the physical space10 was used. The filter width (Δ) was equal to the cubic root of the grid cell volume (i.e., equal to the characteristic length of the grid cell). The filtering operation filters out the turbulent structures with length scales smaller than the filter width. Thus, the resulting equations govern the dynamics of the large-scale structures. However, because of the nonlinear nature of the conservation equations, the filtering process gives rise to unknown subgrid scale (SGS) terms that have to be modeled.10 The unknown terms arising from the filtering operation applied to the momentum equation and the energy equation are the SGS stress tensor and the SGS heat flux, respectively. The LES Favre-filtered (i.e., mass-weighted filtered) c-equation can be written as ∂F~c þ ∇ 3 ðF~u~cÞ þ ∇ 3 ½Fð~uc
~u~cÞ ¼ ∇ 3 ðFD∇cÞ þ ω_ c ∂t

ð2Þ

where the overbar (—) denotes a filtered quantity and the tilde (∼) denotes a Favre-filtered quantity. In eq 2, the (three) unknown terms are the SGS reaction progress variable flux (third term on the left-hand side), the filtered molecular diffusion (first term on the right-hand side), and the SGS reaction rate (second term on the right-hand side).

2.1. Closures for Subgrid Scale (SGS) Stress Tensor and Scalar Fluxes. The SGS stress tensor was described using the

dynamic Smagorinsky
Lilly eddy viscosity model.15 The model coefficient was dynamically calculated during the LES computations using the information about the local instantaneous flow conditions provided by the smallest scales of the resolved (known) field. This allowed the resulting eddy viscosity to respond properly to the local flow structures. The closure for the SGS fluxes of heat and reaction progress variable was achieved through the gradient hypothesis.10 The SGS turbulent Prandtl and Schmidt numbers were obtained by applying the dynamic procedure used for the SGS stress tensor to the SGS scalar fluxes. 2.2. Subgrid Scale (SGS) Combustion Model. To handle the flame
turbulence interaction, the Flame Surface Density (FSD) formalism was used.10 Accordingly, the two right-hand side terms in eq 2 (i.e., the filtered terms of molecular diffusion and reaction rate) were included in a single flame front displacement term, Fw|∇c|, which was expressed as Fwj∇cj ¼ ∇ 3 ðFD∇cÞ þ ω_ c ¼ ÆFwæs Σ

ð3Þ

where Σ is the SGS flame surface density (i.e., the SGS flame surface per unit volume) and ÆFwæs is the surface-averaged massweighted displacement speed, which was approximated by the product of the fresh gas density (F0) and the laminar burning velocity (SL).16 Σ was expressed as a function of the SGS flame wrinkling factor, ΞΔ (i.e., the SGS flame surface divided by its projection in the propagating direction), which takes into account the coupling of flame propagation and unresolved turbulence: Fwj∇cj ¼ ÆFwæs Σ ¼ F0 SL ΞΔ j∇cj ̅

ð4Þ

ΞΔ in eq 4 was modeled according to the SGS combustion closure by Charlette et al.:17
 Δ β ΞΔ ¼ 1 þ ηc ( " ! 0 # )β 0 Δ Δ uΔ u ¼ 1 þ min ,Γ , , ReΔ Δ ð5Þ δF δF SL SL In eq 5, ΞΔ is written in terms of a power-law expression involving an inner cutoff scale (ηc), an outer cutoff scale (i.e., the filter scale) (Δ), and the parameter β as an exponent. The inner cutoff length scale ηc, defined as the inverse mean curvature of the flame, limits wrinkling at the smallest length scales of the flame. ηc was modeled by introducing an efficiency function (Γ), which takes into account the net straining effect of all relevant turbulent scales smaller than the filter size Δ. A spectral analysis of the DNS results of elementary flame
vortex interactions by Colin et al.18 was performed to construct Γ as a function of the filter scale-to-laminar flame thickness ratio (Δ/δF), the SGS turbulent velocity-to-laminar burning velocity ratio (u0 Δ/SL), and the SGS turbulent Reynolds number (ReΔ). Furthermore, Γ was corrected such that the eddies whose characteristic speed falls below SL/2 (very slow eddies) do not wrinkle the flame. The presence of the min operator in eq 5 is due to the fact that the model could predict inner cutoff scales smaller than the laminar flame thickness, δF. To avoid this problem (i.e., to keep ηc g δF), the expression was clipped at the laminar flame thickness. 7705

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Figure 1. Schematic representations of (a) the test rig by Long et al.13 (not to scale) and (b) the flame
vortex interaction.

Table 1. Number of Grid Cells and Average Wall-Clock Time Per Iteration for the 40-mm and 20-mm Orifices Value 40-mm-diameter

20-mm-diameter

parameter

orifice

orifice

number of grid cells

7 819 240

6 383 198

average wall-clock time

32.03 s

28.56 s

per iterationa a

Computations run using the SGS combustion model by Charlette et al.17

Equation 5 is essentially a flame wrinkling model that, however, also works well beyond the wrinkled regime of combustion.6,17 In order to quantify the sensitivity of LES to the presence of the SGS combustion model, computations were also run by setting ΞΔ = 1 in eq 4 (i.e., without the SGS combustion model). 2.3. Configuration and Conditions. LES computations were run for the experiments carried out by Long et al.,13 using the test rig schematized in Figure 1a. Within this rig, a quiescent premixed charge of stoichiometric methane and air was ignited inside a small cylindrical prechamber (length = 35 mm, diameter = 70 mm) linked to the main chamber (150 mm  150 mm  150 mm) via a small orifice (length of 30 mm; variable diameter). The bottom end of the prechamber was fully closed. The upper end of the main chamber was sealed by a thin poly(vinyl chloride) (PVC) membrane that ruptured soon after ignition, allowing the exhaust gases to escape. Ignition was provided at the center of the bottom end of the prechamber. After ignition, the flame propagated through the prechamber, pushing unburned charge ahead of the flame front through the orifice. This motion of the reactants through the constriction resulted in a toroidal vortex being shed into the main chamber. As the flame continued to propagate through the charge, it interacted with the vortex structure, distorting the flame and altering its burning velocity. Figure 1b shows a twodimensional sketch of the flame
vortex interaction (the toroidal vortex was schematized as a vortex pair). The nature (i.e., size and velocity) of the vortices produced in the main chamber is strongly dependent on the orifice diameter.13 In this work, large eddy simulations were performed for two different orifice diameters: 40 mm and 20 mm. 2.4. Numerical Solution. The LES model equations were discretized using a finite-volume formulation on three-dimensional nonuniform unstructured grids with a cell characteristic

length Δ, varying in the range of 0.5
1 mm. The grids were refined in the exit zone of the orifices, where the flame
vortex interaction occurs. Table 1 gives the number of hexahedral cells used for the 40-mm and 20-mm orifices. The values of the percentage of the resolved turbulent kinetic energy were in the range of 80
95% for both grids, thus satisfying Pope’s criterion.9 Bounded central differences were used to discretize convective terms and second-order central differences to calculate diffusion terms. Second-order implicit time integration was used to discretize temporal derivatives. Adiabatic boundary conditions were applied at the walls (bottom, lateral, and top faces of the prechamber; lateral face of the orifice; bottom and vertical faces of the main chamber). Boundary layer effects were taken into account using a blended linear/logarithmic law-of-the-wall.19 Outside the main chamber, the computational domain was extended to simulate the presence of a dump vessel. At the boundaries of this additional domain, the static pressure was imposed as equal to the atmospheric pressure (initial pressure). The initial conditions had energy and reaction progress variable set to zero everywhere. The initial velocity field was quiescent. Ignition was obtained by means of a hemispherical patch, with a radius equal to 1 mm, of hot combustion products at the center of the bottom end of the prechamber. Parallel computations were run by means of the segregated pressure-based solver of the ANSYS Fluent (release 6.3.26) code.20 The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) modified for unsteady simulations21 was used to implement the pressure-correction equation. The solution for each time step required ∼8 iterations to converge with the residual of each equation that was 11 ms in Figure 8a and t > 11.66 ms in Figure 8b. During stage 1, the interaction with the flow structures only wrinkles the flame front. This stage of wrinkled flame propagation is essentially the same in Figure 8a as it is in Figure 8b. During the transition stage (stage 2), the interaction becomes stronger and a flame pocket starts to leave the main front. The fate of this pocket is dependent on the SGS reaction rate. When using the SGS combustion model (Figure 8a), the SGS reaction rate is sufficiently high to allow the pocket formed to survive (without quenching) the interaction with the flow. The pocket

Figure 7. Time sequence of the instantaneous LES maps of the SGS reaction rate obtained (a) using the SGS combustion model by Charlette et al.17 and (b) without the SGS combustion model: 40-mm orifice (central section of the computational domain). 7709

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Figure 8. Time sequence of the instantaneous LES maps of the SGS reaction rate obtained (a) using the SGS combustion model by Charlette et al.17 and (b) without the SGS combustion model: 20-mm orifice (central section of the computational domain).

Figure 9. Flame speed versus the axial distance from the ignition face as obtained from LES computations run using the SGS combustion model by Charlette et al.17 and without the SGS combustion model: 40-mm and 20-mm orifices.

then grows and merges with the main flame front, which is thereby enriched. Conversely, in the absence of the SGS combustion model (Figure 8b), the SGS reaction rate is too low and the pocket quenches. On the other hand, the lower SGS reaction rate also makes the flame front more susceptible to fragmentation. Before the first pocket is fully extinguished, further pockets

of different sizes are formed. The entrainment action of the large vortices allows these pockets to reconnect with the main front. However, such reconnection seems to be rather weak and, thus, unable to sustain the flame development with the same strength as in Figure 8a. During stage 3, pockets are by now merged and/or extinguished: the recompacted flame front propagates in a pseudosteady manner, dominating the flow field. This is more evident in Figure 8a than in Figure 8b. Global assessment of Figure 8 then shows that the differences between the solutions obtained with and without the SGS combustion model result from the different evolution of stage 2 and, in particular, from the different capability of the pockets formed during this stage to reinforce the propagation of the main flame front. The key role of stage 2 is further substantiated by Figure 9, which shows, for both orifices, the flame speed versus the axial distance from the ignition face, as obtained from LES computations run using the SGS combustion model by Charlette et al.17 and without the SGS combustion model. The flame speed was evaluated from the displacement of the maximum downstream location of the flame front. The black rectangle along the x-axis indicates the position of the exit section of the orifice (65 mm). The vertical solid lines represent the boundaries between the various stages of flame propagation downstream of the 20-mm orifice (stages 1, 2, and 3). Regardless of the presence of the SGS combustion model, both orifices exhibit similar trends with steps of flame acceleration past the orifice, deceleration (i.e., flame 7710

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Industrial & Engineering Chemistry Research expansion) into the orifice wake, and reacceleration toward the vented end. However, faster flame propagation is found when using the SGS combustion model, especially in the case of the 20-mm orifice. In particular, it is interesting to observe in Figure 9 that, for the 40-mm orifice (i.e., for the wrinkled regime), the two flame speed profiles almost overlap. Indeed, little deviations between the values of the flame speed predicted using the SGS combustion model and those obtained without the SGS combustion model are also found during the wrinkled flame propagation (stage 1) downstream of the 20-mm orifice. In contrast, as the flame progresses through stages 2 and 3, which also involve the effects of flame pocket
main front interactions, deviations increase. This highlights the need to take into account the SGS flame
turbulence interaction in the LES model to reproduce the flame propagation in the breakthrough regime (i.e., in a more intense turbulent combustion regime) correctly.

4. SUMMARY AND CONCLUSIONS In this paper, the sensitivity of large eddy simulation (LES) to the presence of the combustion submodel was investigated for transient interactions between premixed flame fronts and toroidal vortex structures generated at the wake of a circular orifice. To this end, LES computations were run, with and without the combustion submodel, for two orifice diameters: 40 mm and 20 mm. Nonuniform unstructured grids with a cell characteristic length varying in the range of 0.5
1 mm were used. In going from the 40-mm orifice to the 20-mm orifice, both the size and velocity of the vortex increase, leading to a different regime of interaction with the flame: the vortex only wrinkles the flame front in the 40-mm case (wrinkled regime) and also disrupts the continuity of the front, giving rise to the formation of separate reaction zones (i.e., flame pockets that leave the main front), in the 20-mm case (breakthrough regime). It has been found that the impact of the combustion submodel on LES predictions is strongly dependent on the regime of interaction. Regardless of the presence of the combustion submodel, results for the 40-mm orifice are substantially the same in both qualitative (shape and structure of the flame front) and quantitative (flame speed) terms. Conversely, at the wake of the 20-mm orifice, the intensity of the flame
vortex interaction is such that the combustion submodel is strictly needed to reproduce qualitative and quantitative characteristics of the flame propagation correctly. In the absence of the combustion submodel, the lower reaction rate compromises the capability of the pockets formed to grow and merge with the main flame front, thus sustaining its development. Unrealistic quenching phenomena are indeed observed for such pockets, which cause slower flame propagation. The continuously growing power of parallel computers will enable large eddy simulations of turbulent premixed combustion to be performed with fine grid resolution (of the order of the laminar flame thickness), such as that used in the present work, also at geometry scales larger than laboratory scales. In view of this, the main implication of the results obtained here is that LES modeling efforts should be focused on the development of combustion closures able to catch the transition through more intense regimes of premixed flame
vortex interaction than the wrinkled regime for which LES can be seen as a fully predictive tool. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: +39 0817622673. Fax: +39 0817622915. E-mail: valeria. [email protected].

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’ ACKNOWLEDGMENT This research was partially performed within the framework of Accordo di Programma per l’Attivita di Ricerca di Sistema MSECNR, Progetto: Gas Naturale/Biocombustibili. Numerical simulations were also run on the HPC cluster Matrix of the Italian InterUniversity Consortium for the Application of Super-Computing for Universities and Research (CASPUR, www.caspur.it), within the framework of Standard HPC Grant 2011. The authors gratefully acknowledge the support received by the Consortium. ’ DEDICATION This paper is dedicated to our mentor, Prof. Gennaro Russo, who has always pushed for the development of the current High Performance Computing Laboratory at the Istituto di Ricerche sulla Combustione. Thanks to his constant encouragement, helpful criticism, and efficient guidance, we have acquired the competences needed to study complex phenomena of flame
turbulence interaction by means of advanced CFD modeling and simulation. We shall always be indebted to him for this. ’ NOMENCLATURE c = reaction progress variable D = diffusion coefficient Rf = flame radius Rv = vortex radius ReΔ = subgrid scale turbulent Reynolds number SL = laminar burning velocity t = time u = velocity u0 Δ = subgrid scale turbulent velocity uθ,max = maximum vortex rotational velocity w = local flame front displacement speed relative to the flow Greek Symbols

β = exponent of the subgrid scale combustion model17 Γ = efficiency function Δ = filter width (equal to the characteristic length of the grid cell in this work) δF = laminar flame thickness ηc = inner cutoff flame scale ΞΔ = subgrid scale flame wrinkling factor F = density Fo = unburned gas density Σ = subgrid scale flame surface density ω·c = reaction rate

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