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A flame surface density approach to large-eddy simulation of premixed turbulent combustion
Proceedings of the Combustion Institute, Volume 28, 2000/pp. 51–58
A FLAME SURFACE DENSITY APPROACH TO LARGE-EDDY SIMULATION OF PREMIXED TURBULENT COMBUSTION E. R. HAWKES and R. S. CANT CFD Laboratory Cambridge University Engineering Department Trumpington Street Cambridge CB2 1PZ, UK
The flame surface density approach to the modeling of premixed turbulent combustion is well established in the context of Reynolds-averaged simulations. For the future, it is necessary to consider large-eddy simulation (LES), which is likely to offer major advantages in terms of physical accuracy, particularly for unsteady combustion problems. LES relies on spatial filtering for the removal of unresolved phenomena whose characteristic length scales are smaller than the computational grid scale. Thus, there is a need for soundly based physical modeling at the subgrid scales. The aim of this paper is to explore the usefulness of the flame surface density concept as a basis for LES modeling of premixed turbulent combustion. A transport equation for the filtered flame surface density is presented, and models are proposed for unclosed terms. Comparison with Reynolds-averaged modeling is shown to reveal some interesting similarities and differences. These were exploited together with known physics and statistical results from experiment and from direct numerical stimulation in order to gain insight and refine the modeling. The model has been implemented in a combustion LES code together with standard models for scalar and momentum transport. Computational results were obtained for a simple three-dimensional flame propagation test problem, and the relative importance of contributing terms in the modeled equation for flame surface density was assessed. Straining and curvature are shown to have a major influence at both the resolved and subgrid levels.
The LES filtering operation results in unclosed terms, at the subgrid scale (SGS), that represent interactions between the small- and large-scale flow structures. Assuming low Mach number, single-step chemistry, and unit Lewis number, the thermochemistry of the problem may be represented by a single progress variable, c, which takes the value zero in the unburned reactants and unity in the fully burned products. The filtered equation for c is
Introduction Common approaches to the simulation of turbulent combustion include direct numerical simulation (DNS). Reynolds-averaged Navier Stokes (RANS), and large eddy simulation (LES). DNS involves complete resolution of the reacting flow field, down to the smallest physical scales. In the RANS case, the turbulent motion and turbulent fluctuations of the scalar fields are not resolved at all, and moment closure methods are used to model the effects of the turbulent fluctuations on the mean fields. In contrast, LES partially resolves the turbulent motion and scalar fluctuations. The LES approach involves the solution of a spatially filtered system of equations to determine the large-scale structures of the flow. Only variations of flow quantities on a scale larger than the filter size, D, are resolved. For industrial situations, D may be determined by computational limitations. It will usually lie well above the resolution required for DNS, and below the turbulence integral scale. LES potentially offers advantages over other methods in the simulation of turbulent combustion. The approach is inherently time dependent due to the need to simulate the largescale motion and is therefore particularly well suited to unsteady applications, for example, dynamic oscillations in a gas-turbine combustor.
q¯ c˜ q¯ u˜j c˜ Ⳮ Ⳮ q¯ (uj c ⳮ u˜jc˜) t xj xj ¯˙ Ⳮ qD c ⳱x (1) xj xj ˙ is the reaction rate. where D is the diffusivity, and x LES filtering is denoted by (. . .), and (. . .) denotes a density-weighted LES filtering operation [1]. This work is concerned with the modeling of the unclosed reaction rate and molecular diffusion terms. A model is also required for the subgrid scalar transport term, q( ¯ uj c ⳮ u˜jc˜). This aspect of the modeling is to be addressed in future work, and for now, a simple gradient-transport approach has been adopted. Most SGS modeling efforts have been concerned with the transport of momentum [1], and comparatively little effort has been applied to the issues of
51
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
reaction rate modeling and scalar transport. These problems are more challenging in that there are important interactions between reaction, diffusion, and convection even at the smallest physical scales [2], which are much smaller than D. This is fundamentally different from momentum transport, where the influence of the small scales on the large ones is mainly dissipative. Also, the momentum field exhibits a tendency to return to isotropy at the small scales. The scalar field may not do so at the same length or timescale, and in the presence of a flame, it may not become isotropic at all. A further complication is the effect of non-gradient transport (NGT) of scalars, due to pressure effects linked to the expansion associated with heat release. NGT (often called counter-gradient transport) has been observed in the context of LES modeling both experimentally [3] and in LES-filtered results from DNS [4]. Approaches to modeling the reaction rate term in LES of premixed combustion have included the use of an artificially thickened flame [5], a G-equation [6], and an eddy break-up model [7]. New modeling possibilities are presented with the assumption that the chemical reaction takes place in thin propagating surfaces that locally resemble laminar flames. Such surfaces are strained and wrinkled by the embedding turbulence and are commonly referred to as laminar flamelets. Flamelet approaches, most notably the Bray-Moss-Libby formalism [8], have been widely employed and have proven robust in practice [9,10] over a broad range of applications. The flamelet approach taken in this work is based on the concept of the subgrid flame surface density, Rsg, which is essentially the flame surface area per unit volume contained within the LES filtering volume. This concept has also been explored by Charlette et al. [11] and by Boger et al. [4] and offers significant opportunities for the inclusion of accurate physical modeling at the LES subgrid scale. An alternative formulation based on a flame wrinkling factor has also been presented by Weller et al. [12]. A great deal of research has been devoted to the RANS equivalent, R (see, for example, Refs. [13–16]). The present work is concerned with the formulation and modeling of a transport equation for flame surface density (FSD) within the framework of LES for premixed turbulent combustion. A definition of FSD is presented and discussed, and the corresponding modeled transport equation is compared and contrasted with previous work in this area. The similarities and differences between the LES formulation and the well-established RANS approach are examined, and the partial resolution available in LES is exploited to obtain submodels representing key physical effects. Computational results are presented for a series of flame propagation test cases, and conclusions are drawn regarding the importance of individual terms within the model.
FSD Modeling The LES flame surface density is defined as Rsg ⳱ d(c ⳮ c*)|c|, which contrasts with the RANS approach: R ⳱ d(c ⳮ c*)|c|, where denotes a conventional averaging operation. An LES-weighted flame surface average is defined for a quantity Q, as (Q)s ⳱ Qd(c ⳮ c*)|c|/Rsg. The unclosed reaction and diffusion terms in the c˜ transport equation may then be written as ¯˙ Ⳮ (qDc) ⳱ qw|c| ⳱ x
1
冮 (qw) R
sgdc*
s
0
(2)
where w is the local flame propagation speed. For very thin flames, (qw)s and Rsg are independent of the choice of c*, and 1
冮
0
(qw)s Rsgdc* (qw)s Rsg
For practical purposes, (qw)s may be approximated by qruL, where qr is the fresh gas density and uL is the laminar flame speed. The problem remains to determine Rsg. In Ref. [11], an algebraic model for Rsg was devised by balancing subgrid production and dissipation terms in a modeled transport equation. In many practical situations, including those with significant unsteadiness, the transport equation may prove more useful in exploiting knowledge of the physical mechanisms of flame production and propagation to achieve a more realistic estimate of Rsg. This applies particularly in a situation in which the flame is partially resolved and mean terms can be included. Following a procedure similar to that of Pope [13], the following unclosed transport equation may be derived for Rsg: Rsg Ⳮ (u˜iRsg) Ⳮ ((ui)s ⳮ u˜i)Rsg) ⳱ (aT)s t xi xi Rsg ⳮ
N ((wNi)s Rsg) Ⳮ w k xi xk
冢
冣
Rsg
s
The local flame normal direction is denoted by N, where N ⳱ ⳮc/|c|. The quantity (aT)s represents the straining effect of the fluid motion on the flame surface and is given by
冢
(aT)s ⳱ (dij ⳮ NiNj)
ui xj
冣
s
Modeling of Transport Equation Modeling of the transport equation for R in the context of RANS has led to a number of different approaches and to detailed comparisons between them [14,18]. The main difference between RANS and LES modeling of FSD is that LES can exploit the resolved propagation and strain effects. As far as
FLAME SURFACE DENSITY APPROACH TO LES
possible, LES should give correct results regardless of the choice of the filter size, D. Therefore, the following restrictions were placed on the present model. First, the FSD transport equation should revert to the exact closed equation as the flow becomes fully resolved. Full resolution is interpreted here to mean that subgrid fluctuations of velocity are negligible, and D is much smaller than the radius of curvature of the laminar flame surface. This occurs only for a nearly planar flame or as D goes to zero, which corresponds to a DNS. In this situation, Rsg |c¯| for a very thin flame, and provided there is a good estimate of (qw)s, the progress variable equation is also closed. The second restriction is that if D is large, the model should reflect accepted RANS models for flame in which turbulent fluctuations are completely unresolved. This reflects the approach of Speziale [19] to SGS modeling of momentum transport. Terms in the transport equation may be identified with physical effects. Physical and realizability arguments, supplemented with statistical information from DNS and experimental results, may be used to devise appropriate models. Effects of Propagation and Curvature The terms ⳮ • ((wN)sRsg) Ⳮ (w • N)sRsg represent the combined effects of propagation and curvature of the flame surface. The first term in this expression represents effects of planar propagation, and the second term represents production or destruction of FSD due to the combined effects of curvature and propagation. Under a strict flamelet assumption that the c field consists only of fresh reactants and fully burned products, it may be shown that (N)s ⳱ ⳮc¯/Rsg. reflecting the relation derived by Cant et al. [15] for RANS simulations. The unweighted filtered progress variable c¯ may be related to its densityweighted equivalent c¯ using (1 Ⳮ s)c˜ c¯ ⳱ 1 Ⳮ sc˜
(3)
where s is the heat-release parameter. It is interesting to note that this expression corresponds directly to a relation obtained within the Bray-Moss-Libby formalism [8] for application to RANS modelling. If it is assumed that the propagation speed, w, is constant at the subgrid level, then the split between resolved and subgrid scales may be expressed as ⳮ • ((wN)s Rsg) Ⳮ (w • N)s Rsg ⳱ Pmean Ⳮ Psg where Pmean ⳱ ⳮ • ((w)s (N)s Rsg) Ⳮ (w)s • (N)s Rsg Psg ⳱ ⳮ(w)s ( • (N)s ⳮ ( • N)s)Rsg
53
The term Pmean represents the resolved component and may be seen to revert to the fully resolved expression as D → 0. The term Psg represents the difference between the effect of the resolved curvature term and the actual net effect of the curvature term. It vanishes as the scalar field becomes fully resolved and becomes more important with decreasing resolution. From continuity considerations on a one-dimensional flame, (w)s may be estimated as uL(1 Ⳮ sc*). Some models have included a mean planar propagation effect [15–18,20], but most (except Cant et al. in [15]) have neglected the mean destruction and production terms associated with curvature. Also, Veynante et al. [3] have shown that mean propagation is not negligible as usually assumed. In RANS simulations, the equivalent of Psg has traditionally been modeled as a destruction term proportional to R2sg [14]. Trouve´ and Poinsot [21] showed (for RANS) that the curvature term acts as a production term on the fresh gas side and a dissipation term on the burned gas side of the flame brush. This effect is included in the present model through the mean curvature term. In most RANS models, the modeled terms in general do not vanish for a resolved (laminar) flow, and therefore these models cannot be used directly in LES. An exception is the model of Cant et al. [15], in which a realizability analysis of a one-dimensional flame was used to show that the net destruction of FSD must vanish as ␣ → 0, where ␣ ⳱ 1 ⳮ |Ns|2. In Ref. [15], ␣ is referred to as a resolution factor, since the condition ␣ ⳱ 0 equates to complete resolution. The model for Psg used in this work resembles the commonly employed coherent flame model (CFM) [14] but includes the resolution factor ␣ in order to make the model vanish as the flow becomes resolved. Psg ⳱ ⳮ␣buL
兺2sg
(1 ⳮ c˜)
where b is a model constant. Straining Term The term (aT)s represents the straining effect of the surrounding fluid on the flame surface and may be decomposed as Smean ⳱ (dij ⳮ nij) Ssg ⳱ (aT)S ⳮ Smean
u˜i xj ⳮ Shr
where nij ⳱ (Ni)s(Nj)s Ⳮ 1/3␣dij, and Shr is a term that accounts for the effects of heat release. Due to expansion across the flame, the velocity gradients at the flame surface will be different from the velocity
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
gradients of the mean flow. This term is discussed in the next section. Many studies have neglected the mean strain term, but Veynante et al. [22] have shown experimentally that it should not be neglected for RANS simulations. With increased resolution of the turbulent motion, LES needs to exploit this term. The above expression for the mean strain term was derived for RANS in Ref. [15], on the assumption that fluctuations in N are isotropic. Although this is shown to be inaccurate for RANS in Ref. [22], it is a first step toward representation of Smean. Other estimates, based on the model in Ref. [22] or on the model by Mantel and Borghi [23], do not revert to the exact equations as the flow becomes fully resolved and therefore may not be applicable to LES. The unresolved strain term is usually modeled as proportional to the inverse of a turbulent timescale. The simplest approach is to scale the term with the strain rate at the LES filter cut-off scale, 冪k/D. Nevertheless, models based solely on turbulence timescales do not account for any chemical or molecular transport effects in the response of the flame to the imposed strain rate. Differently sized vortices have a different effect in terms of straining the flame, depending on the relative sizes of the vortices compared with the laminar flame thickness, dL, and the characteristic velocities of the vortices relative to the propagation speed, w. One model that does attempt to reproduce these effects is the intermittent turbulence net flame stretch (ITNFS) model [24]. The straining term is multiplied by an efficiency function, CK, that depends on both the relative velocity u⬘/uL and the relative length scale D/dL. The velocity fluctuation magnitude u⬘ is taken as u⬘ ⳱ 2/3冪k, where k is the subgrid turbulence kinetic energy. The ITNFS model is adopted in the present formulation because it is the only currently available approach that incorporates such a dependence on the local character of the flame. Thus, the subgrid straining term employed here is Ssg ⳱ CK(u⬘/uL, D/dL)冪k/D Turbulent Transport The term (ui)⬙s ⳱ ((ui)s ⳮ u˜i) may be interpreted as a subgrid flux of FSD. This term is strongly connected with the subgrid flux of progress variable uc ⳮ u˜c˜. Cant et al. [15] have derived a theoretical relation between the RANS equivalents of these terms for the case of an infinitely thin flame, where it can be assumed that the fluctuations of the surface normal are uncorrelated with the fluctuations in velocity. Also, Bidaux and Bray [25] have derived an expression relating the two terms based on the assumption of a linear variation of velocity across the flamefront and relying on the results of DNS [26].
The form of this expression shows that if the scalar flux of progress variable is controlled mainly by nongradient mechanisms, then so too is the flux of FSD. NGT has been observed for the progress variable flux both experimentally ([3,27]) and in DNS ([4]), and therefore NGT must be expected in the FSD flux. It is expected to be most dominant when turbulence levels are low and there is a large expansion associated with heat release. Not only is NGT required on physical grounds, but in LES of heat-releasing flames, it is necessary to ensure realistic results. Considering the case of a planar flame with heat release, it may be seen that all terms in the equations vanish except for the mean propagation and subgrid transport terms. The effect of the mean propagation term is to propagate the whole Rsg profile relative to the mean flow with a constant speed (w)s. Since the flow velocity varies through the flame, this will result in a net propagation speed that is faster at the front of the flame than at the back, which results in an unphysical thickening of the flame. The surface fluctuation velocity is modeled as a contribution due to turbulent fluctuations in velocity and a NGT contribution due to heat release that is connected with the slip velocity between reactants and products, u¯p ⳮ u¯r. To establish the form for the NGT contribution, it is instructive to consider the simple case of planar laminar propagation. It may be seen that in the absence of turbulent and pressure gradient effects, |u¯p ⳮ u¯r| suL. However, for curved flames, the magnitude of the slip velocity may reduce below suL. This is due to the fact that the normals are no longer all aligned in the same direction, reducing the mean expansion effect. As the flame becomes more wrinkled, two competing effects are at play. On one hand, the flame propagates faster, resulting in a greater slip velocity, and on the other, the normals become less aligned, resulting in reduced slip velocity. The first effect may be modeled through the surface averaged normal, (N)s, which reduces in magnitude as the flame becomes more wrinkled at the subgrid level. The second effect is more difficult to model, and work is in progress to deal adequately with this term. For now, the following model is employed for the slip velocities: u¯p ⳮ u¯r ⳮsuL (N)s
(4)
Heat release also affects the mean production term as well as the mean planar propagation term, and this is expressed through the term Shr, which was introduced in the previous section. This term is modeled using an adjustment to the resolved curvature effect. In summary, the following models are applied for turbulent transport and production due to heat release:
FLAME SURFACE DENSITY APPROACH TO LES
55
Fig. 1. Resolved flame surface: the surface c˜ ⳱ 0.5 at time t ⳱ 0.75.
(ui)⬙s ⳱ ⳮ(K ⳮ c˜)suL (Ni)s ⳮ
mT 1 Rsg ScR Rsg xi
Shr ⳱ ⳮ(K ⳮ c˜)suL • (N)s Rsg where mT is the turbulent eddy viscosity for LES, and ScR is a subgrid Schmidt number for FSD. The constant K depends on the isosurface chosen to define the flame and is taken to be equal to c*, making the model equations independent of c*. The model for (ui)⬙s is similar to the model suggested by Veynante et al. [26], but differs in that here the efficiency of NGT is reduced with increased turbulence, instead of increasing the efficiency of the gradient diffusion term. The model reduces to the exact equations for the case of a fully resolved, one-dimensional, incompressible, unit Lewis number flame. Flame Propagation Test Case The transport equations for Rsg and c˜ are implemented using a spatially and temporally second-order compressible finite-volume code for the case of a statistically one-dimensional propagating flame. The domain considered is a three-dimensional rectangular box, with Navier-Stokes characteristic boundary conditions [28] applied at the inflow and outflow boundaries, and all other boundaries periodic. The turbulent inflow condition is implemented by generating a random velocity field in Fourier space. The generated field satisfies continuity, and a physically realistic energy spectrum is imposed [29]. The outflow condition corresponds to the partially
non-reflecting condition of [28]. SGS momentum transport was modeled using a standard gradient transport formulation based on the SGS kinetic energy, k [30]. The inflow condition for k was calculated from the known spectrum, assuming a spectral cut-off LES filter. The turbulent flame is initialized from a planar laminar flame solution and is allowed to evolve in the oncoming turbulent field. Parameters for the simulation are summarized as follows, with appropriate normalization using the length of the domain together with a nominal velocity scale: uL ⳱ 0.2; dL ⳱ 0.005; s ⳱ 2.3; b ⳱ 1.0; Mach number, M ⳱ 0.125; Reynolds number, Re ⳱ 1000; total inflow turbulence kinetic energy, ktot ⳱ 0.12; eddy viscosity constant, Cm ⳱ 0.089; subgrid Schmidt numbers, Scc ⳱ ScR ⳱ 1.0. A surface c* ⳱ 0Ⳮ is used to define the location of the flame, giving (w)s uL and K ⳱ 0. The staggered spatial grid consisted of 64 ⳯ 64 ⳯ 64 equally spaced nodes, with D ⳱ 6.4 grid cells. Figure 1 shows the isosurface c˜ ⳱ 0.5 for the time t ⳱ 0.75. Propagation is in the downward direction. Although this situation is statistically one-dimensional, it may be seen that the LES resolves the large three-dimensional coherent structures of the curved flame surface. It is instructive to note that a RANS simulation of the same problem would be strictly one-dimensional. Figure 2 illustrates a slice through the simulated flame at time t ⳱ 0.5, showing contours of constant progress variable together with the local intensity of Rsg. Propagation is downward, and the intensity of Rsg is represented by a scale running from zero
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
Fig. 2. Flame surface: a slice through the flame, showing contours of c˜ and intensity of Rsg.
Fig. 3. FSD source terms: source terms in the modeled transport equation for Rsg, averaged in the homogeneous directions.
(blue) to maximum (deep red). It is clear that the resolved flame brush remains well-structured, and that there is significant variation of Rsg along the flame that is related to the local flame geometry. Figure 3 shows a budget of source terms calculated from the Rsg transport equation for time t ⳱ 0.5 and averaged in the homogeneous directions (which are normal to the direction of mean propagation). It is clear that the largest influences arise from the subgrid curvature and convection terms, and that there is a net imbalance that corresponds to the time rate of change of Rsg. The resolved strain term, Smean, acts mainly as a production term throughout the flame, and it may be observed that Smean and Ssg have approximately the same order of magnitude. Thus it is clear that Smean should not be neglected even in this statistically one-dimensional
Fig. 4. Resolved curvature source: source term in modeled transport equation for Rsg, corresponding to the effect of resolved curvature, and contours of c˜. Blue indicates FSD destruction, red indicates FSD production.
problem in which the mean velocity gradient is aligned with the mean flame normal. In more general situations, when the two are not aligned, this term is expected to be even more significant. It may be observed from Fig. 3 that the resolved curvature and resolved propagation source terms are of similar magnitude and are comparable in size to the strain source terms. Therefore, where the planar propagation term is significant and there is significant resolved curvature of the flame, the curvature term also needs to be included. This term has been neglected by almost all other models, (except for that of Cant et al. [15]). Also of note is the countergradient (NGT) term, which is again of a similar magnitude to the other terms. Most FSD models have not included a term to account for this effect. Also in Fig. 3, it may be seen that the resolved propagation source term acts as expected, causing an increase in FSD on the fresh gas side and a decrease on the burned gas side of the flame. The behavior of the resolved curvature term is displayed in greater detail in Fig. 4, where it is clear that this term acts as a production term in regions where the flame is convex to the reactants, and as a destruction term where the flame is concave to the reactants. It is interesting to note the strong destruction of FSD along a path behind the each of the cusp points. Comparison of Figs. 2 and 4 serves to illustrate the strongly dynamic nature of the flame propagation process, in that the existence of large values of Rsg is the result of many contributions to production and destruction taken over a significant time period. This underlines the need for models based on the solution of a full transport equation for Rsg. Conclusions An FSD approach to LES of premixed turbulent combustion has been presented. A modeled transport equation has been developed based on physical
FLAME SURFACE DENSITY APPROACH TO LES
arguments, on the results of experiment and of DNS, and on previous work in the context of RANS modeling. The modeled transport equation differs from RANS models for FSD in that mean production terms associated with effects of propagation, curvature, and strain from the resolved flow field are included in the equations. Also, effects of NGT have been included through an adjustment to the mean propagation and production terms that accounts for expansion through the flame. The transport equation reverts to the exact equations for a propagating flame surface as the scalar and velocity fields become resolved (as the filtering size becomes much smaller than the flame radius of curvature). At high levels of subgrid wrinkling, the model reflects accepted RANS models for FSD. Results have been presented for a freely propagating turbulent flame. These showed that the mean curvature FSD source terms were of comparable magnitude to the planar propagation term. Therefore, when uL is comparable to the turbulent propagation rate, the mean curvature term should be included. The resolved and subgrid strain production terms were also of comparable magnitude, and therefore the resolved strain term should also not be neglected as is generally assumed in the literature. Large-eddy simulation is a promising approach to the simulation of turbulent combustion, and the FSD approach looks to be a worthwhile avenue for further research. Work in progress involves model refinements by comparison with LES-filtered DNS results. Model validation against experiment is also planned for the near future. Nomenclature D ;q qr c c* ui g˙v D R⬘ R Rsg X(u, v, t) w uL N aT
LES filter width density reactant density progress variable progress variable defining flame surface fluid velocity reaction rate species diffusivity fine-grained FSD RANS-averaged FSD LES-averaged FSD flame surface point local surface propagation velocity laminar flame speed local flame normal direction strain rate of flame surface due to fluid motion
s ⳱ (Tp ⳮ Tr)/Tr heat-release parameter Pmean mean component of FSD propagation source term Psg SGS component of FSD propagationcurvature source term
␣⳱1ⳮ (N)s • (N)s b Smean Shr Ssg k CK u⬘ (ui)⬙s ur, up F K u|s mt ScR Scsg M ktot Cm Re dL
57
resolution factor model parameter for Psg mean component of FSD strain source component of strain source connect with heat release subgrid component of strain source subgrid kinetic energy efficiency function of ITNFS model RANS fluctuation velocity surface fluctuation velocity conditional LES reactant and product averages LES filter function model parameter fluid velocity at flame surface subgrid eddy viscosity schmidt number for FSD progress variable subgrid Schmidt number Mach number total inflow turbulent kinetic energy eddy viscosity constant Reynolds number laminar flame thickness Acknowledgments
The financial support of Alstom Gas Turbine Ltd. is gratefully acknowledged. REFERENCES 1. Rogallo, R. S., and Moin, P., Annu. Rev. Fluid Mech. 16:99–137 (1984). 2. Menon, S., McMurtry, P. A., and Kerstein, A. R., in Large Eddy Simulation of Complex Engineering and Geophysical Flows, Cambridge University Press, Cambridge, 1993, pp. 287–314. 3. Veynante, D., Piana, J., Duclos, J. M., and Martel, C., Proc. Combust. Inst. 26:413–420 (1996). 4. Boger, M., Veynante, D., Boughanem, H., and Trouvee´, A., Proc. Combust. Inst. 27:917–925 (1998). 5. Thibaut, D., and Candel, S., Combust. Flame 113:53– 65 (1998). 6. Menon, S., and Jou W.-H., Combust. Sci. Technol. 75:53–72 (1991). 7. Fureby, C., and Moller, S.-I., AIAA J. 33(12):2339– 2347 (1995). 8. Bray, K. N. C., Libby, P. A., and Moss, J. B., Combust. Flame 61:87–102 (1985). 9. Bray, K. N. C., and Peters, N., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, eds.), Academic Press Limited, London, 1994, pp. 63–113. 10. Poinsot, T., Veynante, D., and Candel, S., Proc. Combust. Inst. 23:613–619 (1990). 11. Charlette, F., Trouve´, A., Boger, M., and Veynante, D., “A Flame Surface Density Model for Large Eddy Simulations of Turbulent Premixed Flames,” paper no.
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12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
OA2, in Proceedings of the Joint Meeting of the British, German and French Sections of the Combustion Institute, The Combustion Institute, Nancy, May 1999. Weller, H. G., Tabor, G., Gosman, A. D., and Fureby, C., Proc. Combust. Inst. 27:899–907 (1998). Pope, S., Int. J. Eng. Sci. 26(5):445–469 (1988). Duclos, J. M., Veynante, D., and Poinsot, T., Combust. Flame 95:101–117 (1993). Cant, R. S., Pope, S. B., and Bray, K. N. C., Proc. Combust. Inst. 23:809–815 (1990). Candel, S. M., and Poinsot T. J., Combust. Sci. Technol. 70:1–15 (1990). Vervisch, L., Bidaux, E., Bray, K. N. C., and Kollmann, W., Phys. Fluids A 7(10):2496–2503 (1995). Prasad, R. O. S., and Gore, J. P., Combust. Flame 116:1–14 (1999). Speziale, C., AIAA J. 36(2):173–184 (1998). Wu, A. S., and Bray, K. N. C., Combust. Flame 109:43– 64 (1997). Trouve´, A., and Poinsot, T., J. Fluid Mech. 278:1–31 (1994).
22. Veynante, D., Duclos, J. M., and Piana, J., Proc. Combust. Inst. 25:1249–1256 (1990). 23. Mantel, T., and Borghi, R., Combust. Flame 96(4):443– 457 (1994). 24. Meneveau, C., and Poinsot, T., Combust. Flame 86:311–332 (1991). 25. Bidaux, E., and Bray, K. N. C., unpublished data, 1994. 26. Veynante, D., Trouve´, A., Bray, K. N. C., and Mantel, T., J. Fluid Mech. 332:263–293 (1997). 27. Moss, J. B., Combust. Sci. Technol. 22:119–129 (1980). 28. Poinsot, T., and Lele, S. K., J. Comput. Phys. 101:104– 129 (1992). 29. Lee, M. J., and Reynolds, W. C., “Numerical Experiments on the Structure of Homogenous Turbulence,” technical report TF-24, Thermosciences Division, Dept. of Mechanical Engineering, Stanford University, Stanford, 1985. 30. Yoshizawa, A., Ninth Symposium on Turbulent Shear Flows, Kyoto, Japan, 1993.
COMMENTS J. Chomiak, Chalmers University of Technology, Sweden. There are several flame surface density combustion models known from the literature. They differ strongly on the level of constants used to determine the effects of turbulence on flame/surface generation and destruction. In LES, the subgrid turbulence effects on the aforementioned have to be estimated, and the estimate must depend on the resolution. How do you handle this problem? Author’s Reply. No author reply. ● A. Trove, George Washington University, USA. One of the difficulties found in the filtered reaction progress variable approach is that the reaction progress variable has to remain smooth on the computational grid. This means that the ratio of filter size divided by grid size is above 1. Can you tell us what the minimum value of this ratio is in your simulations and comment on the associated computational cost?
Author’s Reply. As pointed out in the comment, all approaches based on the filtered progress variable approach suffer from the problem that the reaction progress variable needs to remain smooth on the computational grid. It is expected that for a planar laminar flame in which the filter size is greater than the laminar flame thickness that the filtered progress variable profile will have a thickness of the order of the LES filter width due to the smoothing action of the filtering operation. This is a problem for these approaches in that a large ratio of filter size to grid width is required, of the order of ten to one, which compares to the value of two to one commonly employed in cold flow simulations. However, for wrinkled flames, the filtered profile (and thus the simulated flame thickness) may have a thickness that is greater than the filter width. This relaxed the computational requirement somewhat. In the simulations described in the present work, a ratio of 6.4 grid cells to one filter width is employed, however, there are a larger number of points, approximately 10, within the simulated flame profile.
A FLAME SURFACE DENSITY APPROACH TO LARGE-EDDY SIMULATION OF PREMIXED TURBULENT COMBUSTION E. R. HAWKES and R. S. CANT CFD Laboratory Cambridge University Engineering Department Trumpington Street Cambridge CB2 1PZ, UK
The flame surface density approach to the modeling of premixed turbulent combustion is well established in the context of Reynolds-averaged simulations. For the future, it is necessary to consider large-eddy simulation (LES), which is likely to offer major advantages in terms of physical accuracy, particularly for unsteady combustion problems. LES relies on spatial filtering for the removal of unresolved phenomena whose characteristic length scales are smaller than the computational grid scale. Thus, there is a need for soundly based physical modeling at the subgrid scales. The aim of this paper is to explore the usefulness of the flame surface density concept as a basis for LES modeling of premixed turbulent combustion. A transport equation for the filtered flame surface density is presented, and models are proposed for unclosed terms. Comparison with Reynolds-averaged modeling is shown to reveal some interesting similarities and differences. These were exploited together with known physics and statistical results from experiment and from direct numerical stimulation in order to gain insight and refine the modeling. The model has been implemented in a combustion LES code together with standard models for scalar and momentum transport. Computational results were obtained for a simple three-dimensional flame propagation test problem, and the relative importance of contributing terms in the modeled equation for flame surface density was assessed. Straining and curvature are shown to have a major influence at both the resolved and subgrid levels.
The LES filtering operation results in unclosed terms, at the subgrid scale (SGS), that represent interactions between the small- and large-scale flow structures. Assuming low Mach number, single-step chemistry, and unit Lewis number, the thermochemistry of the problem may be represented by a single progress variable, c, which takes the value zero in the unburned reactants and unity in the fully burned products. The filtered equation for c is
Introduction Common approaches to the simulation of turbulent combustion include direct numerical simulation (DNS). Reynolds-averaged Navier Stokes (RANS), and large eddy simulation (LES). DNS involves complete resolution of the reacting flow field, down to the smallest physical scales. In the RANS case, the turbulent motion and turbulent fluctuations of the scalar fields are not resolved at all, and moment closure methods are used to model the effects of the turbulent fluctuations on the mean fields. In contrast, LES partially resolves the turbulent motion and scalar fluctuations. The LES approach involves the solution of a spatially filtered system of equations to determine the large-scale structures of the flow. Only variations of flow quantities on a scale larger than the filter size, D, are resolved. For industrial situations, D may be determined by computational limitations. It will usually lie well above the resolution required for DNS, and below the turbulence integral scale. LES potentially offers advantages over other methods in the simulation of turbulent combustion. The approach is inherently time dependent due to the need to simulate the largescale motion and is therefore particularly well suited to unsteady applications, for example, dynamic oscillations in a gas-turbine combustor.
q¯ c˜ q¯ u˜j c˜ Ⳮ Ⳮ q¯ (uj c ⳮ u˜jc˜) t xj xj ¯˙ Ⳮ qD c ⳱x (1) xj xj ˙ is the reaction rate. where D is the diffusivity, and x LES filtering is denoted by (. . .), and (. . .) denotes a density-weighted LES filtering operation [1]. This work is concerned with the modeling of the unclosed reaction rate and molecular diffusion terms. A model is also required for the subgrid scalar transport term, q( ¯ uj c ⳮ u˜jc˜). This aspect of the modeling is to be addressed in future work, and for now, a simple gradient-transport approach has been adopted. Most SGS modeling efforts have been concerned with the transport of momentum [1], and comparatively little effort has been applied to the issues of
51
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
reaction rate modeling and scalar transport. These problems are more challenging in that there are important interactions between reaction, diffusion, and convection even at the smallest physical scales [2], which are much smaller than D. This is fundamentally different from momentum transport, where the influence of the small scales on the large ones is mainly dissipative. Also, the momentum field exhibits a tendency to return to isotropy at the small scales. The scalar field may not do so at the same length or timescale, and in the presence of a flame, it may not become isotropic at all. A further complication is the effect of non-gradient transport (NGT) of scalars, due to pressure effects linked to the expansion associated with heat release. NGT (often called counter-gradient transport) has been observed in the context of LES modeling both experimentally [3] and in LES-filtered results from DNS [4]. Approaches to modeling the reaction rate term in LES of premixed combustion have included the use of an artificially thickened flame [5], a G-equation [6], and an eddy break-up model [7]. New modeling possibilities are presented with the assumption that the chemical reaction takes place in thin propagating surfaces that locally resemble laminar flames. Such surfaces are strained and wrinkled by the embedding turbulence and are commonly referred to as laminar flamelets. Flamelet approaches, most notably the Bray-Moss-Libby formalism [8], have been widely employed and have proven robust in practice [9,10] over a broad range of applications. The flamelet approach taken in this work is based on the concept of the subgrid flame surface density, Rsg, which is essentially the flame surface area per unit volume contained within the LES filtering volume. This concept has also been explored by Charlette et al. [11] and by Boger et al. [4] and offers significant opportunities for the inclusion of accurate physical modeling at the LES subgrid scale. An alternative formulation based on a flame wrinkling factor has also been presented by Weller et al. [12]. A great deal of research has been devoted to the RANS equivalent, R (see, for example, Refs. [13–16]). The present work is concerned with the formulation and modeling of a transport equation for flame surface density (FSD) within the framework of LES for premixed turbulent combustion. A definition of FSD is presented and discussed, and the corresponding modeled transport equation is compared and contrasted with previous work in this area. The similarities and differences between the LES formulation and the well-established RANS approach are examined, and the partial resolution available in LES is exploited to obtain submodels representing key physical effects. Computational results are presented for a series of flame propagation test cases, and conclusions are drawn regarding the importance of individual terms within the model.
FSD Modeling The LES flame surface density is defined as Rsg ⳱ d(c ⳮ c*)|c|, which contrasts with the RANS approach: R ⳱ d(c ⳮ c*)|c|, where denotes a conventional averaging operation. An LES-weighted flame surface average is defined for a quantity Q, as (Q)s ⳱ Qd(c ⳮ c*)|c|/Rsg. The unclosed reaction and diffusion terms in the c˜ transport equation may then be written as ¯˙ Ⳮ (qDc) ⳱ qw|c| ⳱ x
1
冮 (qw) R
sgdc*
s
0
(2)
where w is the local flame propagation speed. For very thin flames, (qw)s and Rsg are independent of the choice of c*, and 1
冮
0
(qw)s Rsgdc* (qw)s Rsg
For practical purposes, (qw)s may be approximated by qruL, where qr is the fresh gas density and uL is the laminar flame speed. The problem remains to determine Rsg. In Ref. [11], an algebraic model for Rsg was devised by balancing subgrid production and dissipation terms in a modeled transport equation. In many practical situations, including those with significant unsteadiness, the transport equation may prove more useful in exploiting knowledge of the physical mechanisms of flame production and propagation to achieve a more realistic estimate of Rsg. This applies particularly in a situation in which the flame is partially resolved and mean terms can be included. Following a procedure similar to that of Pope [13], the following unclosed transport equation may be derived for Rsg: Rsg Ⳮ (u˜iRsg) Ⳮ ((ui)s ⳮ u˜i)Rsg) ⳱ (aT)s t xi xi Rsg ⳮ
N ((wNi)s Rsg) Ⳮ w k xi xk
冢
冣
Rsg
s
The local flame normal direction is denoted by N, where N ⳱ ⳮc/|c|. The quantity (aT)s represents the straining effect of the fluid motion on the flame surface and is given by
冢
(aT)s ⳱ (dij ⳮ NiNj)
ui xj
冣
s
Modeling of Transport Equation Modeling of the transport equation for R in the context of RANS has led to a number of different approaches and to detailed comparisons between them [14,18]. The main difference between RANS and LES modeling of FSD is that LES can exploit the resolved propagation and strain effects. As far as
FLAME SURFACE DENSITY APPROACH TO LES
possible, LES should give correct results regardless of the choice of the filter size, D. Therefore, the following restrictions were placed on the present model. First, the FSD transport equation should revert to the exact closed equation as the flow becomes fully resolved. Full resolution is interpreted here to mean that subgrid fluctuations of velocity are negligible, and D is much smaller than the radius of curvature of the laminar flame surface. This occurs only for a nearly planar flame or as D goes to zero, which corresponds to a DNS. In this situation, Rsg |c¯| for a very thin flame, and provided there is a good estimate of (qw)s, the progress variable equation is also closed. The second restriction is that if D is large, the model should reflect accepted RANS models for flame in which turbulent fluctuations are completely unresolved. This reflects the approach of Speziale [19] to SGS modeling of momentum transport. Terms in the transport equation may be identified with physical effects. Physical and realizability arguments, supplemented with statistical information from DNS and experimental results, may be used to devise appropriate models. Effects of Propagation and Curvature The terms ⳮ • ((wN)sRsg) Ⳮ (w • N)sRsg represent the combined effects of propagation and curvature of the flame surface. The first term in this expression represents effects of planar propagation, and the second term represents production or destruction of FSD due to the combined effects of curvature and propagation. Under a strict flamelet assumption that the c field consists only of fresh reactants and fully burned products, it may be shown that (N)s ⳱ ⳮc¯/Rsg. reflecting the relation derived by Cant et al. [15] for RANS simulations. The unweighted filtered progress variable c¯ may be related to its densityweighted equivalent c¯ using (1 Ⳮ s)c˜ c¯ ⳱ 1 Ⳮ sc˜
(3)
where s is the heat-release parameter. It is interesting to note that this expression corresponds directly to a relation obtained within the Bray-Moss-Libby formalism [8] for application to RANS modelling. If it is assumed that the propagation speed, w, is constant at the subgrid level, then the split between resolved and subgrid scales may be expressed as ⳮ • ((wN)s Rsg) Ⳮ (w • N)s Rsg ⳱ Pmean Ⳮ Psg where Pmean ⳱ ⳮ • ((w)s (N)s Rsg) Ⳮ (w)s • (N)s Rsg Psg ⳱ ⳮ(w)s ( • (N)s ⳮ ( • N)s)Rsg
53
The term Pmean represents the resolved component and may be seen to revert to the fully resolved expression as D → 0. The term Psg represents the difference between the effect of the resolved curvature term and the actual net effect of the curvature term. It vanishes as the scalar field becomes fully resolved and becomes more important with decreasing resolution. From continuity considerations on a one-dimensional flame, (w)s may be estimated as uL(1 Ⳮ sc*). Some models have included a mean planar propagation effect [15–18,20], but most (except Cant et al. in [15]) have neglected the mean destruction and production terms associated with curvature. Also, Veynante et al. [3] have shown that mean propagation is not negligible as usually assumed. In RANS simulations, the equivalent of Psg has traditionally been modeled as a destruction term proportional to R2sg [14]. Trouve´ and Poinsot [21] showed (for RANS) that the curvature term acts as a production term on the fresh gas side and a dissipation term on the burned gas side of the flame brush. This effect is included in the present model through the mean curvature term. In most RANS models, the modeled terms in general do not vanish for a resolved (laminar) flow, and therefore these models cannot be used directly in LES. An exception is the model of Cant et al. [15], in which a realizability analysis of a one-dimensional flame was used to show that the net destruction of FSD must vanish as ␣ → 0, where ␣ ⳱ 1 ⳮ |Ns|2. In Ref. [15], ␣ is referred to as a resolution factor, since the condition ␣ ⳱ 0 equates to complete resolution. The model for Psg used in this work resembles the commonly employed coherent flame model (CFM) [14] but includes the resolution factor ␣ in order to make the model vanish as the flow becomes resolved. Psg ⳱ ⳮ␣buL
兺2sg
(1 ⳮ c˜)
where b is a model constant. Straining Term The term (aT)s represents the straining effect of the surrounding fluid on the flame surface and may be decomposed as Smean ⳱ (dij ⳮ nij) Ssg ⳱ (aT)S ⳮ Smean
u˜i xj ⳮ Shr
where nij ⳱ (Ni)s(Nj)s Ⳮ 1/3␣dij, and Shr is a term that accounts for the effects of heat release. Due to expansion across the flame, the velocity gradients at the flame surface will be different from the velocity
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
gradients of the mean flow. This term is discussed in the next section. Many studies have neglected the mean strain term, but Veynante et al. [22] have shown experimentally that it should not be neglected for RANS simulations. With increased resolution of the turbulent motion, LES needs to exploit this term. The above expression for the mean strain term was derived for RANS in Ref. [15], on the assumption that fluctuations in N are isotropic. Although this is shown to be inaccurate for RANS in Ref. [22], it is a first step toward representation of Smean. Other estimates, based on the model in Ref. [22] or on the model by Mantel and Borghi [23], do not revert to the exact equations as the flow becomes fully resolved and therefore may not be applicable to LES. The unresolved strain term is usually modeled as proportional to the inverse of a turbulent timescale. The simplest approach is to scale the term with the strain rate at the LES filter cut-off scale, 冪k/D. Nevertheless, models based solely on turbulence timescales do not account for any chemical or molecular transport effects in the response of the flame to the imposed strain rate. Differently sized vortices have a different effect in terms of straining the flame, depending on the relative sizes of the vortices compared with the laminar flame thickness, dL, and the characteristic velocities of the vortices relative to the propagation speed, w. One model that does attempt to reproduce these effects is the intermittent turbulence net flame stretch (ITNFS) model [24]. The straining term is multiplied by an efficiency function, CK, that depends on both the relative velocity u⬘/uL and the relative length scale D/dL. The velocity fluctuation magnitude u⬘ is taken as u⬘ ⳱ 2/3冪k, where k is the subgrid turbulence kinetic energy. The ITNFS model is adopted in the present formulation because it is the only currently available approach that incorporates such a dependence on the local character of the flame. Thus, the subgrid straining term employed here is Ssg ⳱ CK(u⬘/uL, D/dL)冪k/D Turbulent Transport The term (ui)⬙s ⳱ ((ui)s ⳮ u˜i) may be interpreted as a subgrid flux of FSD. This term is strongly connected with the subgrid flux of progress variable uc ⳮ u˜c˜. Cant et al. [15] have derived a theoretical relation between the RANS equivalents of these terms for the case of an infinitely thin flame, where it can be assumed that the fluctuations of the surface normal are uncorrelated with the fluctuations in velocity. Also, Bidaux and Bray [25] have derived an expression relating the two terms based on the assumption of a linear variation of velocity across the flamefront and relying on the results of DNS [26].
The form of this expression shows that if the scalar flux of progress variable is controlled mainly by nongradient mechanisms, then so too is the flux of FSD. NGT has been observed for the progress variable flux both experimentally ([3,27]) and in DNS ([4]), and therefore NGT must be expected in the FSD flux. It is expected to be most dominant when turbulence levels are low and there is a large expansion associated with heat release. Not only is NGT required on physical grounds, but in LES of heat-releasing flames, it is necessary to ensure realistic results. Considering the case of a planar flame with heat release, it may be seen that all terms in the equations vanish except for the mean propagation and subgrid transport terms. The effect of the mean propagation term is to propagate the whole Rsg profile relative to the mean flow with a constant speed (w)s. Since the flow velocity varies through the flame, this will result in a net propagation speed that is faster at the front of the flame than at the back, which results in an unphysical thickening of the flame. The surface fluctuation velocity is modeled as a contribution due to turbulent fluctuations in velocity and a NGT contribution due to heat release that is connected with the slip velocity between reactants and products, u¯p ⳮ u¯r. To establish the form for the NGT contribution, it is instructive to consider the simple case of planar laminar propagation. It may be seen that in the absence of turbulent and pressure gradient effects, |u¯p ⳮ u¯r| suL. However, for curved flames, the magnitude of the slip velocity may reduce below suL. This is due to the fact that the normals are no longer all aligned in the same direction, reducing the mean expansion effect. As the flame becomes more wrinkled, two competing effects are at play. On one hand, the flame propagates faster, resulting in a greater slip velocity, and on the other, the normals become less aligned, resulting in reduced slip velocity. The first effect may be modeled through the surface averaged normal, (N)s, which reduces in magnitude as the flame becomes more wrinkled at the subgrid level. The second effect is more difficult to model, and work is in progress to deal adequately with this term. For now, the following model is employed for the slip velocities: u¯p ⳮ u¯r ⳮsuL (N)s
(4)
Heat release also affects the mean production term as well as the mean planar propagation term, and this is expressed through the term Shr, which was introduced in the previous section. This term is modeled using an adjustment to the resolved curvature effect. In summary, the following models are applied for turbulent transport and production due to heat release:
FLAME SURFACE DENSITY APPROACH TO LES
55
Fig. 1. Resolved flame surface: the surface c˜ ⳱ 0.5 at time t ⳱ 0.75.
(ui)⬙s ⳱ ⳮ(K ⳮ c˜)suL (Ni)s ⳮ
mT 1 Rsg ScR Rsg xi
Shr ⳱ ⳮ(K ⳮ c˜)suL • (N)s Rsg where mT is the turbulent eddy viscosity for LES, and ScR is a subgrid Schmidt number for FSD. The constant K depends on the isosurface chosen to define the flame and is taken to be equal to c*, making the model equations independent of c*. The model for (ui)⬙s is similar to the model suggested by Veynante et al. [26], but differs in that here the efficiency of NGT is reduced with increased turbulence, instead of increasing the efficiency of the gradient diffusion term. The model reduces to the exact equations for the case of a fully resolved, one-dimensional, incompressible, unit Lewis number flame. Flame Propagation Test Case The transport equations for Rsg and c˜ are implemented using a spatially and temporally second-order compressible finite-volume code for the case of a statistically one-dimensional propagating flame. The domain considered is a three-dimensional rectangular box, with Navier-Stokes characteristic boundary conditions [28] applied at the inflow and outflow boundaries, and all other boundaries periodic. The turbulent inflow condition is implemented by generating a random velocity field in Fourier space. The generated field satisfies continuity, and a physically realistic energy spectrum is imposed [29]. The outflow condition corresponds to the partially
non-reflecting condition of [28]. SGS momentum transport was modeled using a standard gradient transport formulation based on the SGS kinetic energy, k [30]. The inflow condition for k was calculated from the known spectrum, assuming a spectral cut-off LES filter. The turbulent flame is initialized from a planar laminar flame solution and is allowed to evolve in the oncoming turbulent field. Parameters for the simulation are summarized as follows, with appropriate normalization using the length of the domain together with a nominal velocity scale: uL ⳱ 0.2; dL ⳱ 0.005; s ⳱ 2.3; b ⳱ 1.0; Mach number, M ⳱ 0.125; Reynolds number, Re ⳱ 1000; total inflow turbulence kinetic energy, ktot ⳱ 0.12; eddy viscosity constant, Cm ⳱ 0.089; subgrid Schmidt numbers, Scc ⳱ ScR ⳱ 1.0. A surface c* ⳱ 0Ⳮ is used to define the location of the flame, giving (w)s uL and K ⳱ 0. The staggered spatial grid consisted of 64 ⳯ 64 ⳯ 64 equally spaced nodes, with D ⳱ 6.4 grid cells. Figure 1 shows the isosurface c˜ ⳱ 0.5 for the time t ⳱ 0.75. Propagation is in the downward direction. Although this situation is statistically one-dimensional, it may be seen that the LES resolves the large three-dimensional coherent structures of the curved flame surface. It is instructive to note that a RANS simulation of the same problem would be strictly one-dimensional. Figure 2 illustrates a slice through the simulated flame at time t ⳱ 0.5, showing contours of constant progress variable together with the local intensity of Rsg. Propagation is downward, and the intensity of Rsg is represented by a scale running from zero
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
Fig. 2. Flame surface: a slice through the flame, showing contours of c˜ and intensity of Rsg.
Fig. 3. FSD source terms: source terms in the modeled transport equation for Rsg, averaged in the homogeneous directions.
(blue) to maximum (deep red). It is clear that the resolved flame brush remains well-structured, and that there is significant variation of Rsg along the flame that is related to the local flame geometry. Figure 3 shows a budget of source terms calculated from the Rsg transport equation for time t ⳱ 0.5 and averaged in the homogeneous directions (which are normal to the direction of mean propagation). It is clear that the largest influences arise from the subgrid curvature and convection terms, and that there is a net imbalance that corresponds to the time rate of change of Rsg. The resolved strain term, Smean, acts mainly as a production term throughout the flame, and it may be observed that Smean and Ssg have approximately the same order of magnitude. Thus it is clear that Smean should not be neglected even in this statistically one-dimensional
Fig. 4. Resolved curvature source: source term in modeled transport equation for Rsg, corresponding to the effect of resolved curvature, and contours of c˜. Blue indicates FSD destruction, red indicates FSD production.
problem in which the mean velocity gradient is aligned with the mean flame normal. In more general situations, when the two are not aligned, this term is expected to be even more significant. It may be observed from Fig. 3 that the resolved curvature and resolved propagation source terms are of similar magnitude and are comparable in size to the strain source terms. Therefore, where the planar propagation term is significant and there is significant resolved curvature of the flame, the curvature term also needs to be included. This term has been neglected by almost all other models, (except for that of Cant et al. [15]). Also of note is the countergradient (NGT) term, which is again of a similar magnitude to the other terms. Most FSD models have not included a term to account for this effect. Also in Fig. 3, it may be seen that the resolved propagation source term acts as expected, causing an increase in FSD on the fresh gas side and a decrease on the burned gas side of the flame. The behavior of the resolved curvature term is displayed in greater detail in Fig. 4, where it is clear that this term acts as a production term in regions where the flame is convex to the reactants, and as a destruction term where the flame is concave to the reactants. It is interesting to note the strong destruction of FSD along a path behind the each of the cusp points. Comparison of Figs. 2 and 4 serves to illustrate the strongly dynamic nature of the flame propagation process, in that the existence of large values of Rsg is the result of many contributions to production and destruction taken over a significant time period. This underlines the need for models based on the solution of a full transport equation for Rsg. Conclusions An FSD approach to LES of premixed turbulent combustion has been presented. A modeled transport equation has been developed based on physical
FLAME SURFACE DENSITY APPROACH TO LES
arguments, on the results of experiment and of DNS, and on previous work in the context of RANS modeling. The modeled transport equation differs from RANS models for FSD in that mean production terms associated with effects of propagation, curvature, and strain from the resolved flow field are included in the equations. Also, effects of NGT have been included through an adjustment to the mean propagation and production terms that accounts for expansion through the flame. The transport equation reverts to the exact equations for a propagating flame surface as the scalar and velocity fields become resolved (as the filtering size becomes much smaller than the flame radius of curvature). At high levels of subgrid wrinkling, the model reflects accepted RANS models for FSD. Results have been presented for a freely propagating turbulent flame. These showed that the mean curvature FSD source terms were of comparable magnitude to the planar propagation term. Therefore, when uL is comparable to the turbulent propagation rate, the mean curvature term should be included. The resolved and subgrid strain production terms were also of comparable magnitude, and therefore the resolved strain term should also not be neglected as is generally assumed in the literature. Large-eddy simulation is a promising approach to the simulation of turbulent combustion, and the FSD approach looks to be a worthwhile avenue for further research. Work in progress involves model refinements by comparison with LES-filtered DNS results. Model validation against experiment is also planned for the near future. Nomenclature D ;q qr c c* ui g˙v D R⬘ R Rsg X(u, v, t) w uL N aT
LES filter width density reactant density progress variable progress variable defining flame surface fluid velocity reaction rate species diffusivity fine-grained FSD RANS-averaged FSD LES-averaged FSD flame surface point local surface propagation velocity laminar flame speed local flame normal direction strain rate of flame surface due to fluid motion
s ⳱ (Tp ⳮ Tr)/Tr heat-release parameter Pmean mean component of FSD propagation source term Psg SGS component of FSD propagationcurvature source term
␣⳱1ⳮ (N)s • (N)s b Smean Shr Ssg k CK u⬘ (ui)⬙s ur, up F K u|s mt ScR Scsg M ktot Cm Re dL
57
resolution factor model parameter for Psg mean component of FSD strain source component of strain source connect with heat release subgrid component of strain source subgrid kinetic energy efficiency function of ITNFS model RANS fluctuation velocity surface fluctuation velocity conditional LES reactant and product averages LES filter function model parameter fluid velocity at flame surface subgrid eddy viscosity schmidt number for FSD progress variable subgrid Schmidt number Mach number total inflow turbulent kinetic energy eddy viscosity constant Reynolds number laminar flame thickness Acknowledgments
The financial support of Alstom Gas Turbine Ltd. is gratefully acknowledged. REFERENCES 1. Rogallo, R. S., and Moin, P., Annu. Rev. Fluid Mech. 16:99–137 (1984). 2. Menon, S., McMurtry, P. A., and Kerstein, A. R., in Large Eddy Simulation of Complex Engineering and Geophysical Flows, Cambridge University Press, Cambridge, 1993, pp. 287–314. 3. Veynante, D., Piana, J., Duclos, J. M., and Martel, C., Proc. Combust. Inst. 26:413–420 (1996). 4. Boger, M., Veynante, D., Boughanem, H., and Trouvee´, A., Proc. Combust. Inst. 27:917–925 (1998). 5. Thibaut, D., and Candel, S., Combust. Flame 113:53– 65 (1998). 6. Menon, S., and Jou W.-H., Combust. Sci. Technol. 75:53–72 (1991). 7. Fureby, C., and Moller, S.-I., AIAA J. 33(12):2339– 2347 (1995). 8. Bray, K. N. C., Libby, P. A., and Moss, J. B., Combust. Flame 61:87–102 (1985). 9. Bray, K. N. C., and Peters, N., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, eds.), Academic Press Limited, London, 1994, pp. 63–113. 10. Poinsot, T., Veynante, D., and Candel, S., Proc. Combust. Inst. 23:613–619 (1990). 11. Charlette, F., Trouve´, A., Boger, M., and Veynante, D., “A Flame Surface Density Model for Large Eddy Simulations of Turbulent Premixed Flames,” paper no.
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TURBULENT COMBUSTION—Direct Numerical Simulations and Large Eddy Simulations
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COMMENTS J. Chomiak, Chalmers University of Technology, Sweden. There are several flame surface density combustion models known from the literature. They differ strongly on the level of constants used to determine the effects of turbulence on flame/surface generation and destruction. In LES, the subgrid turbulence effects on the aforementioned have to be estimated, and the estimate must depend on the resolution. How do you handle this problem? Author’s Reply. No author reply. ● A. Trove, George Washington University, USA. One of the difficulties found in the filtered reaction progress variable approach is that the reaction progress variable has to remain smooth on the computational grid. This means that the ratio of filter size divided by grid size is above 1. Can you tell us what the minimum value of this ratio is in your simulations and comment on the associated computational cost?
Author’s Reply. As pointed out in the comment, all approaches based on the filtered progress variable approach suffer from the problem that the reaction progress variable needs to remain smooth on the computational grid. It is expected that for a planar laminar flame in which the filter size is greater than the laminar flame thickness that the filtered progress variable profile will have a thickness of the order of the LES filter width due to the smoothing action of the filtering operation. This is a problem for these approaches in that a large ratio of filter size to grid width is required, of the order of ten to one, which compares to the value of two to one commonly employed in cold flow simulations. However, for wrinkled flames, the filtered profile (and thus the simulated flame thickness) may have a thickness that is greater than the filter width. This relaxed the computational requirement somewhat. In the simulations described in the present work, a ratio of 6.4 grid cells to one filter width is employed, however, there are a larger number of points, approximately 10, within the simulated flame profile.