# A Priori Direct Numerical Simulation Modelling of the Curvature Term of the Flame Surface Density Transport Equation for Nonunity Lewis Number Flames in the Context of Large Eddy Simulations

Authors: Mohit Katragadda; Nilanjan Chakraborty
Journal: International Journal of Chemical Engineering
 (2012)

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## Abstract

A Direct Numerical Simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with Lewis numbersLe ranging from 0.34 to 1.2 has been used to analyse the statistical behaviours of the curvature term of the generalised Flame surface Density (FSD) transport equation, in the context of the Large Eddy Simulation (LES). Lewis number is shown to have significant influences on the statistical behaviours of the resolved and sub-grid parts of the FSD curvature term. It has been found that the existing models for the sub-grid curvature term Csg do not capture the qualitative behaviour of this term extracted from the DNS database for flames with Le<<1. The existing models of Csg only predict negative values, whereas the sub-grid curvature term is shown to assume positive values within the flame brush for the Le=0.34 and 0.6 flames. Here the sub-grid curvature terms arising from combined reaction and normal diffusion and tangential diffusion components of displacement speed are individually modelled, and the new model of the sub-grid curvature term has been found to capture Csg extracted from DNS data satisfactorily for all the different Lewis number flames considered here for a wide range of filter widths.


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## Body

## 1. Introduction

Flame Surface Density (FSD) based reaction rate closure is well established in the context of Reynolds Averaged Navier-Stokes (RANS) simulations of turbulent premixed flames [1, 2]. The increased affordability of high performance computing has made Large Eddy Simulation (LES) an alternative simulation tool, where the large-scale physical processes are resolved, but modelling is still required for the subgrid quantities. The FSD-based reaction rate closure has recently been successfully extended for the purpose of LES [3–14]. In LES simulation of premixed combustion, a Favre-filtered reaction progress variable transport equation is solved alongside other filtered conservation equations. The reaction progress variable is defined as c=(YR0-YR)/(YR0-YR∞), where YR is the mass fraction of a suitable reactant and the subscripts 0 and ∞denote the values in the fully unburned and burned gases, respectively. The generalised FSD is defined as Σgen=|∇c|¯ [3–14], where the overbar indicates the LES filtering process. The Favre-filtered reaction progress variable transport equation takes the following form:
(1)ρ-∂c~∂t+ρ-u~j∂c~∂xj=∂∂xj(ρD∂c∂xj)¯+w˙¯-∂∂xj(ρ-(ujc~-u~jc~)),
where Q~=ρQ¯/ρ-indicates the Favre filtered value of a general variable Q, uj is the velocity component in the jth direction, ρ is the density, D is the molecular diffusivity, and w˙¯ is the filtered reaction rate. The first two terms on right hand side of (1) denote the filtered molecular diffusion and reaction rates, respectively, and their combined contribution can be modelled using Σgen in the following manner:
(2)w˙+∇⋅(ρD∇c)¯=(ρSd)¯sΣgen,
where (Q)¯s=Q|∇c|¯/Σgen indicates the surface-weighted filtered value of a general quantity Q and Sd=Dc/Dt/|∇c| is the displacement speed, which denotes the speed at which a given cisosurface moves normal to itself with respect to an initially coincident material surface. The generalised FSD Σgen is an unclosed quantity and is closed either by using an algebraic expression or by solving a modelled transport equation alongside other conservation equations. The algebraic closure is valid when the generation rate of flame surface area remains in equilibrium with its destruction rate, but this assumption is rendered invalid under unsteady conditions (e.g., combustion instabilities). Under unsteady conditions, it is often advantageous to solve a modelled transport equation ofΣgen. The exact transport equation for the generalised FSD Σgen is given as [1, 4–7, 9, 10, 12]:
(3)∂Σgen∂t+∂(u~jΣgen)∂xj∂Σgen∂t=-∂[((ui)¯s-u~i)Σgen]∂xi+((δij-NiNj)∂ui∂xj)¯sΣgen∂Σgen∂t-∂[(SdNi)¯sΣgen]∂xi+(Sd(∂Ni∂xi))¯sΣgen,
where Ni=-(∂c/∂xi)/|∇c| is the ith component of flame normal vector. The terms on the left hand side of (3) denote transient and mean advection effects, respectively. The first three terms on the right hand side of (3) denote the effects of subgrid convection, flame surface area generation due to fluid-dynamic straining, and flame normal propagation, respectively. The last term of (3) describes the production/destruction of Σgen due to flame curvature κm=(∂Ni/∂xi)/2 and thus referred to as the FSD curvature term [4–7, 9, 10, 12]. It has been found in several previous studies [5–7, 9, 14] that the FSD curvature term remains a leading order contributor to the FSD transport for both unity and nonunity Lewis number turbulent premixed combustion. As the curvature term remains a leading order contributor to the FSD transport, the modelling of (Sd∇·N→)¯sΣgen is crucial for the transport equation-based FSD closure. The statistical behaviour of (Sd∇·N→)¯sΣgen is significantly affected by curvature dependence on Sd [9, 10, 12]. Earlier a prioriDirect Numerical Simulation (DNS) analyses [9, 10, 12] showed that existing models for the subgrid curvature term Csg do not adequately capture the qualitative behaviour of this term obtained from DNS data. Moreover, the model parameters for the existing subgrid curvature term Csg models are found to be strong functions of the LES filter width Δ [9, 10, 12].To date, most existing FSD-based models have been proposed for unity Lewis number flames where the differential diffusion of heat and mass has been ignored. The Lewis number is defined as the ratio of thermal diffusivityαT to mass diffusivity D (i.e., Le=αT/D). The effects of Le on the statistical behaviour of the FSD curvature term (Sd∇·N→)¯sΣgen are yet to be analysed in detail, and this paper aims to bridge this gap in the existing literature. It is worth noting that, in a premixed flame, different species have different values of Lewis number. Thus, specifying a global Lewis number Le characterising the whole combustion process is not straightforward. The Lewis number of the deficient reactant is often considered to be the characteristic Le of the combustion process in question [15, 16]. Moreover, several previous studies [16–29] analysed the effects of differential diffusion of heat and mass by modifying the characteristic Lewis number in isolation, and the same procedure has been adopted here. In the present study, a simplified chemistry-based DNS database of statistically planar turbulent premixed flames with global Lewis numbers ranging from 0.34 to 1.2 has been considered to analyse the statistical behaviour of the FSD curvature term (Sd∇·N→)¯sΣgen in the context of LES. In this context, the main objectives of this study are as follows: (1)
to analyse the statistical behaviours of the subgrid FSD curvature term in the context of LES, for flames with different values of Lewis number;(2)
to propose models for different components of the subgrid FSD curvature terms and assess their performances in comparison to the corresponding quantities extracted from DNS data.The rest of the paper will be organised as follows. The necessary mathematical background will be provided in the next section. This will be followed by a brief description of the numerical implementation related to the DNS database. Following this, results will be presented and subsequently discussed. The main findings will be summarised, and conclusions will be drawn in the final section.


## 2. Mathematical Background

The curvature term of the FSD(Sd∇·N→)¯sΣgen is often decomposed in the following manner [4–7, 9, 10, 12]:
(4)(Sd∇⋅N→)¯sΣgen=Cmean+Csg,
where Cmean and Csg are the resolved and subgrid components of the FSD curvature term, respectively. The resolved curvature term Cmeancan be expressed in three different manners [5, 9, 10, 12]:
(5a)Cmean=(Sd)¯s[∂(Ni)¯s∂xi]Σgen(5b)Cmean=(Sd)¯s[∂Mi∂xi]Σgen,(5c)Cmean=(δij-nij)∂[(Sd)¯s(Ni)¯s]∂xjΣgen,
where Mi=-(∂c-/∂xi)/|∇c-| is the ith component of the resolved flame normal vector. It was demonstrated by Chakraborty and Cant [10, 12] that (5a) provides the best option for the resolved curvature term Cmean, as it gives rise to the smallest magnitude of Csg among all the possibilities shown in (5a)–(5c). Equation (5a) was found to perform the best among the three possibilities shown in (5a), (5b), (5c) for this database. This is advantageous from the perspective of efficient modelling of the FSD curvature term (Sd∇·N→)¯sΣgen as most of the modelling uncertainty is associated with Csg. Moreover, (5a) has also been used for the modelling of (Sd∇·N→)¯sΣgen in previous LES simulations [5–7, 13]. For the present analysis (5a), (i.e., Cmean=(Sd)¯s[∂(Ni)¯s/∂xi]Σgen) will be considered for the resolved curvature term Cmean.It is often useful to decompose the flame displacement speedSd=(Dc/Dt)/|∇c|=[w˙+∇·(ρD∇c)]/ρ|∇c| in the following manner for the purpose of modelling the FSD curvature term [9–12, 30, 31]:
(6a)Sd=Sr+Sn+St,(6b)Sr=w˙ρ|∇c|,(6c)Sn=N→⋅∇(ρDN→⋅∇c)ρ|∇c|,(6d)St=-D∇⋅N→=-2Dκm,
where Sr and Sn are the reaction and normal diffusion components of displacement speed and St is the tangential diffusion component of displacement speed. The following expression for Csg can be obtained using (6a)–(6d) and (5a) (i.e., Cmean=(Sd)¯s[∂(Ni)¯s/∂xi]Σgen):
(7)Csg=Csg1+Csg2=(Sd∂Ni∂xi)¯sΣgen-(Sd)¯s∂(Ni)¯s∂xiΣgen,
where
(8a)Csg1=[((Sr+Sn)∂Ni∂xi)¯sΣgen-(Sr+Sn)¯s∂(Ni)¯s∂xiΣgen],(8b)Csg2=-[(D(∂Ni∂xi)2)¯sΣgen-(D∂Ni∂xi)¯s∂(Ni)¯s∂xiΣgen].
Equation (8a) indicates that curvature (κm=∇·N→/2) dependences of (Sr+Sn) and |∇c| significantly influence the statistical behaviour of Csg1. Equation (8b) suggests that Csg2 is expected to assume negative values throughout the flame brush.Hawkes and Cant [6, 7] modified a version of the Coherent Flamelet Model (CFM) by Candel et al. [2] for the purpose of LES as:
(9)Csg=-αNβ1SLΣgen2(1-c-),
where αN=1-(Nk)¯s(Nk)¯s is a resolution parameter which vanishes when the flow is fully resolved and β1 is a model parameter. Hawkes [5] discussed a possibility of modifying a RANS model proposed by Cant et al. [1] for the purpose of LES as:
(10)Csg=-CHSLΣgen2(1-c-),
where CH=αNβ2(1-(1/3)[1-exp(-10(1-c-)k~/ΣgenSLΔ)], A=10.0, uΔ′=2k~/3 is the subgrid turbulent velocity fluctuation, k~=(ρuiui¯-ρ-u~iu~i)/2ρ- is the subgrid kinetic energy, and β2 is a model parameter. Another model of Csg was proposed by Charlette et al. [4]:
(11)Csg=-β3SL(Σgen-|∇c-|)Σgenc-(1-c-),
where β3 is a model parameter. The models given by (9)–(11) (henceforth will be referred to as CSGCFM, CSGCPB, and CSGCHAR, resp.) ensure that Csg vanishes when the flow is fully resolved (i.e., (Nk)¯s(Nk)¯s=1.0 and Σgen=|∇c-|). A priori DNS assessment of the CSGCFM, CSGCPB, and CSGCHAR models and the modelling of Csg1 and Csg2 will be addressed in Section 4 of this paper.


## 3. Numerical Implementation

In principle combustion, DNS should account for both three dimensionality of turbulence and detailed chemical mechanism. However, until recently, most combustion DNS studies were carried out either in two dimensions with detailed chemistry or in three dimensions with simplified chemistry due to the limitation of computer storage capacity. Although it is now possible to carry out three-dimensional DNS with detailed chemistry, they remain extremely expensive (e.g., millions of CPU hours and thousands of processors [32]) and the cost of an extensive parametric analysis based on three-dimensional detailed chemistry-based DNS often becomes prohibitive. As the present analysis concentrates on an extensive parametric variation in terms of Lewis number, the chemical mechanism is simplified here by an Arrhenius-type irreversible single-step chemical reaction (i.e., Reactants→Products) following several previous studies [1–12, 14]. It has been found that the strain rate and curvature dependences of Sd and |∇c| obtained from three-dimensional simplified chemistry DNS [25–27, 33, 34] are found to be qualitatively similar to the corresponding behaviours obtained from detailed chemistry-based DNS simulations [16, 30, 31, 35]. As the statistical behaviours of the FSD curvature term (Sd∇·N→)¯sΣgen are strongly dependent on the curvature dependences of Sd and |∇c|, the results for this analysis are expected to be valid even for detailed chemistry based simulations at least in a qualitative sense without much loss of generality. Several studies [3–7, 9–12] have concentrated on a priori DNS modelling of FSD based on simplified chemistry in the past and the same approach has been adopted here.A compressible three-dimensional DNS code SENGA [36] was used for the simulations where the conservation equations of mass, momentum, energy, and species are solved in nondimensional form. A cubic domain of each side equal to 24.1δth is considered for the present DNS database where δth is the thermal flame thickness, which is defined as δth=(Tad-T0)/Max|∇T^|L, and the subscript L refers to quantities in an unstrained planar laminar flame with Tad, T0, and T^ being the adiabatic flame, unburned gas, and instantaneous gas temperatures, respectively. The computational domain was discretised using a Cartesian grid of 230×230×230 with equal grid spacing in each direction. The grid spacing Δx is determined based on the flame resolution, and about 10 grid points are kept within the thermal flame thickness δth for all the cases considered here. This grid spacing Δx corresponds to 0.73η, where η is the Kolmogorov length scale. The boundaries in the mean flame propagation were taken to be partially nonreflecting and were implemented using the Navier-Stokes Characteristic Boundary Conditions (NSCBC) technique [37]. The boundary conditions in the transverse direction were taken to be periodic. The spatial derivatives for the internal grid points were evaluated using a tenth-order central differencing scheme, and the order of differentiation gradually decreases to a one-sided 2nd order scheme at the partially nonreflecting boundaries. The time advancement was carried out using an explicit low storage third-order Runge-Kutta scheme [38].For the current DNS database, the turbulent velocity field was initialised using a pseudospectral method [39] following the Batchelor-Townsend turbulent kinetic energy spectrum [40]. The flame is initialised using a steady planar unstrained laminar flame solution. The initial values of normalised root mean square (rms) turbulent velocity fluctuation u′/SL, integral length scale to thermal flame thickness ratio l/δth, heat release parameter τ=(Tad-T0)/T0, Damköhler number Da=lSL/u′δth, and Karlovitz number Ka=(u′/SL)3/2(l/δth)-1/2 are listed in Table 1. According to Peters [41], all the cases considered here can be taken to represent the thin reaction zone regime combustion, as Ka remains greater that unity. Standard values are considered for Prandtl number Pr and the Zel’dovich number β (i.e., Pr=0.7 and β=Tac(Tad-T0)Tad2=6.0, where Tac is the activation temperature).Table 1
Initial values of the simulation parameters and non-dimensional numbers relevant to the DNS database.
Case
Le
u
′
/
S
L
l
/
δ
L
l
/
δ
th
δ
th
/
η
τ
R
e
t
Da
Ka
A
0.34
7.5
1.13
2.45
7.32
4.5
47.0
0.33
34.3
B
0.6
7.5
1.76
2.45
7.32
4.5
47.0
0.33
19.40
C
0.8
7.5
2.13
2.45
7.32
4.5
47.0
0.33
14.70
D
1.0
7.5
2.45
2.45
7.32
4.5
47.0
0.33
11.75
E
1.2
7.5
2.72
2.45
7.32
4.5
47.0
0.33
9.80Under decaying turbulence, DNS simulations should be carried out for a simulation timetsim≥max(tf,tc) [42], where tf=l/u′ is the initial eddy turn over time and tc=δth/SL is the chemical time scale. For this database, the statistics were extracted after about three eddy turn over times (i.e., 3tf=3l/u′), which corresponded to one chemical time scale (i.e., tc=δth/SL). This simulation time remains small but comparable to several studies [3, 24, 28, 43–47] which contributed significantly to the FSD-based modelling in the past. The statistics presented in this paper did not change significantly since halfway through the simulation (i.e., 1.5tf=1.5l/u′). The value of u′/SL in the unburned gas ahead of the flame had decayed by 50% of its initial value when the statistics were extracted. By this time, the normalised integral value l/δth had increased to around 1.7 times of its initial value. The values of u′/SL and l/δth at the time statistics were extracted are also representative of the thin reaction zones regime combustion [41]. This DNS database was used extensively earlier for the purpose of RANS modelling [27, 28, 48, 49], and the interested readers are referred to these papers for further details.The DNS data was explicitly LES filtered using a Gaussian filter kernel in physical space for the purpose ofa priori analysis. The filtered quantity Q-(x→,t), is given by
(12)Q-(x→,t)=∫Q(x→-r→)G(r→)dr→,
where G(r→) is the Gaussian filter kernel, which is defined in the following manner:
(13)G(r→)=(6πΔ2)3/2exp⁡⁡(-6r→⋅r→Δ2).The filtered quantities of interest were extracted for filter widthsΔ ranging from 0.4δth to 2.4δth in steps of 0.4δth. These filter sizes are comparable to the range of Δ used in several previous studies [3, 4, 9–12, 14] for a priori DNS analysis and span a useful range of length scales (i.e., from Δ≈0.4δth, where the flame is partially resolved, up to 2.4δth, where the flame becomes fully unresolved and Δ is comparable to the integral length scale).


## 4. Results and Discussion

The instantaneous isosurfaces ofc ranging from 0.01 to 0.99 at tc=δth/SL are shown in Figure 1, which indicates that the flame wrinkling increases with decreasing Lewis number and this tendency is particularly prevalent for the Le≪1 flames due to thermodiffusive instabilities [17–29]. The unburned reactants diffuse into the reaction zone at a faster rate than the rate at which heat diffuses out in the Le<1 flames. This gives rise to simultaneous presence of high temperature and reactant concentration in the reaction zone for the Le<1 flames, which in turn leads to greater burning rate and flame surface area generation in comparison to the unity Lewis number flame. By contrast, heat diffuses faster than the diffusion rate of reactants into the reaction zone in the case of Le>1, which reduces the burning rate and the rate of flame area generation in comparison to the unity Lewis number flame. The increase in burning rate and flame area generation with decreasing Lewis number can be substantiated by the values of normalised turbulent flame speed ST/SL and normalised flame surface area AT/AL which are presented in Table 2. The values of ST/SL have been evaluated by volume integrating the reaction rate w˙ using the expression ST=(1/ρ0AP)∫Vw˙dV, where AP is the projected area of the flame in the direction of mean flame propagation, while the values of AT/AL have been evaluated by volume integrating |∇c| (i.e., ∫V|∇c|dV) under both turbulent and laminar conditions. Table 2 shows that both ST/SL and AT/AL increase with decreasing Lewis number, and this effect is particularly prevalent in the flames with Le < 1 due to the presence of thermodiffusive instabilities [17–29]. The increase in flame wrinkling with decreasing Lewis number is also visually evident from the c isosurfaces presented in Figure 1.Table 2
The effects of Lewis number on normalised turbulent flame speedST/SL and normalised flame surface area AT/AL after 3.0 initial eddy turn over times.
Case
Le
S
T
/
S
L
A
T
/
A
L
A
0.34
13.70
3.93
B
0.6
4.58
2.66
C
0.8
2.53
2.11
D
1.0
1.83
1.84
E
1.2
1.50
1.76Instantaneous isosurfaces ofc ranging from 0.01 to 0.99 at t=3tf=tc for cases (a) Le=0.34; (b) Le=0.6; (c) Le=0.8; (d) Le=1.0; (e) Le=1.2. The domain size is 24.1δth×24.1δth×24.1δth.
(a)
(b)
(c)
(d)
(e)The variations of(Sd∇·N→)¯sΣgen, Cmean, and Csg conditionally averaged in bins of c~ isosurfaces for cases (a)–(e) are shown in Figure 2 for filter widths Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth, where Δm is the DNS grid size. It is evident from Figure 2 that Le significantly affects the statistical behaviours of the curvature terms. The filter widths Δ=8Δm≈0.8δth, and Δ=24Δm≈2.4δth span a useful range of length scales (i.e., from Δ≈0.8δth, where the flame is partially resolved, up to 2.4δth where the flame becomes fully unresolved and Δ is comparable to the integral length scale). In the Le ≪ 1 flames (e.g., cases (a) and (b)), the FSD curvature term (Sd∇·N→)¯sΣgen behaves as a source term for the major part of the flame brush before assuming negative values towards the burned gas side for Δ=8Δm≈0.8δth. For Δ=24Δm≈2.4δth, the FSD curvature term (Sd∇·N→)¯sΣgen acts as a source (sink) term towards the unburned (burned) gas side of the flame brush in the Le≪1 flames. In the case of Le≈1.0 flames (i.e., cases (c)–(e)) the curvature term (Sd∇·N→)¯sΣgen behaves as a sink type term throughout the flame brush for all filter widths. It can be seen from Figure 2 that Cmean acts as a source (sink) term for cases (a)-(b) ((c)–(e)). The magnitude of Cmean (Csg) decreases (increases) with increasing Δ in all cases, and for large filter widths (Sd∇·N→)¯sΣgen is principally made up of Csg. The LES filtering is a convolution process, and the weighted averaging involved in the filtering process leads to a decrease in the magnitude of Cmean with increasing filter width Δ. The flow becomes increasingly unresolved with increasing filter width Δ, and this is reflected in the rise in Csg magnitude with increasing filter width Δ.Variation of(Sd∇·N→)¯sΣgen (solid line), Cmean=(Sd)¯s∂(Ni)¯s/∂xiΣgen (dashed line) and Csg (asterisk line) conditionally averaged in bins of c~ across the flame brush for filter sizes Δ=8Δm≈0.8δth (top row): (a) Le=0.34 (1st column); (b) Le=0.6 (2nd column); (c) Le=0.8 (3rd column); (d) Le=1.0 (4th column); (e) Le=1.2 (5th column) and for filter size Δ=24Δm≈2.4δth (bottom row): (f) Le=0.34 (1st column); (g) Le=0.6 (2nd column); (h) Le=0.8 (3rd column); (i) Le=1.0 (4th column); (j) Le=1.2 (5th column). All terms are ensemble averaged on c~ isosurfaces in Figure 2 and subsequent cases. All the curvature terms in this and subsequent figures are normalised by SL/δth2.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)The resolved curvature termCmean=(Sd)¯s∂(Ni)¯s/∂xiΣgen can be seen to capture the behaviour of the curvature term (Sd∇·N→)¯sΣgen, well at small filter widths (i.e., Δ≤δth) for flames with Le≈1.0 (i.e., cases (c)–(e)). However, the magnitude of Cmean decreases with increasing Δ and it does not capture the behaviour of the FSD curvature term (Sd∇·N→)¯sΣgen for the Le ≪ 1.0 flames (i.e., cases (a) and (b)). The subgrid curvature term, Csg follows the qualitative behaviour of the FSD curvature term (Sd∇·N→)¯sΣgen for all filter widths. The subgrid curvature term Csg almost entirely makes up the FSD curvature term (Sd∇·N→)¯sΣgen for Δ≫δth, and this is especially true for the Le ≪ 1.0 cases (i.e., cases (a) and (b)). It can further be observed from Figure 2 that Csg assumes positive values towards the unburned gas side of the flame brush in the Le ≪ 1 flames (e.g., cases (a) and (b)), whereas the existing models for Csg allow for only negative values (see (9)–(11)). This suggests that new models for Csg are warranted to account for the influences of nonunity Lewis number (i.e., Le≠1.0).In order to be able to model the subgrid curvature termCsg, the decomposition prescribed in (8a)-(8b) has been used here. The variations of ((Sr+Sn)∇·N→)¯sΣgen, -(D(∇·N→)2)¯sΣgen=-4(Dκm2)¯sΣgen, Csg1 and Csg2 conditionally averaged in bins of c~ isosurfaces for cases (a)–(e) are shown in Figure 3 for filter widths Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth. It is evident from Figure 3 that Csg2 remains negative throughout the flame brush for all cases and follows the qualitative behaviour of (-4(Dκm2)¯sΣgen). A comparison between ((Sr+Sn)∇·N→)¯sΣgen and -4(Dκm2)¯sΣgen reveals that -4(Dκm2)¯sΣgen remains the major contributor to (Sd∇·N→)¯sΣgen for all the flames at all values of Δ, which is consistent with the expected behaviour in the thin reaction zones regime [41]. The contribution of ((Sr+Sn)∇·N→)¯sΣgen remains significant for the Le<1 cases (i.e., cases (a), (b) and (c)), but its contribution remains weak in comparison to the magnitude of -4(Dκm2)¯sΣgen in the Le=1.0 and 1.2 flames (i.e., cases (d) and (e)). Figure 3 demonstrates that Csg1 remains close to the magnitude of ((Sr+Sn)∇·N→)¯sΣgen for all Δ for the Le=1.0 flame (i.e., case (d)), indicating that (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen does not play a major role in capturing the behaviour of ((Sr+Sn)∇·N→)¯sΣgen. However, there is a significant difference in magnitudes of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 for small values of Δ (i.e., Δ<δth) in the nonunity Lewis number flames (i.e., cases (a)–(c) and (e)), which indicates that (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen plays a key role for small values of filter width in these flames. For large values of filter width (i.e., Δ≫δth) Csg1 remains the major contributor to ((Sr+Sn)∇·N→)¯sΣgen for all cases considered here, indicating that (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen plays progressively less important role for increasing values of Δ.Variation of((Sr+Sn)∇·N→)¯sΣgen (solidline), -4(Dκm2)¯sΣgen (dashed line), Csg1 (asterisk line), and Csg2 (squared line) conditionally averaged in bins of c~ across the flame brush for Δ=8Δm≈0.8δth (top row): (a) Le=0.34 (1st column); (b) Le=0.6 (2nd column); (c) Le=0.8 (3rd column); (d) Le=1.0 (4th column); (e) Le=1.2 (5th column) and for filter size Δ=24Δm≈2.4δth (bottom row): (f) Le=0.34 (1st column); (g) Le=0.6 (2nd column); (h) Le=0.8 (3rd column); (i) Le=1.0 (4th column); (j) Le=1.2 (5th column).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)Figure3 shows that there is a significant difference between -4(Dκm2)¯sΣgen and Csg2 for all cases for small values of Δ, and the difference between these quantities decreases with increasing Δ. As most of the contribution of -4(Dκm2)¯sΣgen remains unresolved for large values of Δ, the subgrid curvature term Csg2 remains the major contributor to -4(Dκm2)¯sΣgen, indicating that (-(D∂Ni/∂xi)¯s∂(Ni)¯s/∂xiΣgen) plays progressively less important role for increasing values of Δ where the flame is fully unresolved. However, the contribution of (-(D∂Ni/∂xi)¯s∂(Ni)¯s/∂xiΣgen) remains significant for small values of Δ, where the flame is partially resolved. Figure 3 further shows that the order of magnitudes of both Csg1 and Csg2 remains comparable and thus accurate modelling of Csg1 and Csg2 is necessary for precise predictions of Csg.As the range ofκm values obtained on a flame surface increases with increasing flame wrinkling, the magnitude of -4(Dκm2)¯s increases with decreasing Le, which in turn leads to increasing magnitude of -4(Dκm2)¯sΣgen and Csg2 (see Figure 3). The positive contribution of Csg1 overcomes the negative contribution of Csg2 towards the unburned gas side of the flame brush for the Le=0.34 and 0.6 flames (i.e., cases (a) and (b)) and yields a net positive contribution of Csg towards the reactant side of the flame brush (see Figure 2).The statistical behaviours of((Sr+Sn)∇·N→)¯sΣgen and Csg1 depend on the nature of the curvature κm=∇·N→/2 dependences of (Sr+Sn) and |∇c|, and the variation of (κm)¯s across the flame brush. The correlation coefficients for (Sr+Sn)-κm and |∇c|-κm for five different c isosurfaces across the flame brush for all the cases are shown in Figures 4(a) and 4(b), respectively. For all cases, St=-2Dκm remains negatively correlated with κm with a correlation coefficient close to (−1.0). However, Figures 4(a) and 4(b) demonstrate that Le significantly affects the curvature κm dependences of (Sr+Sn) and |∇c|. It can be seen from Figures 4(a) and 4(b) that (Sr+Sn) and |∇c| remain positively (negatively) correlated with κm for the Le<1.0 (Le>1.0) flames, whereas both (Sr+Sn) and |∇c| show weak curvature dependences in the unity Lewis number flame. The positive correlations between (Sr+Sn) and κm, and between |∇c| and κm strengthen with decreasing Le for the Le<1 flames. The physical explanations for the observed influences of Lewis number on the curvature dependence of (Sr+Sn) and |∇c| have been discussed elsewhere [25–27] and thus will not be discussed in this paper.Correlation coefficients between (a)(Sr+Sn) and κm, and between (b) |∇c| and κm correlations on c= 0.1, 0.3, 0.5, 0.7, and 0.9 isosurfaces for cases A–E. Variation of (κm)¯s×δth with c~ across the flame brush for (c) Δ=8Δm≈0.8δth and (d) Δ=24Δm≈2.4δth for cases A–E.
(a)
(b)
(c)
(d)The variations of(κm)¯s conditionally averaged in bins of c~ isosurfaces for cases A–E are shown in Figures 4(c) and 4(d) for filter widths Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth, respectively. It is evident from Figures 4(c) and 4(d) that (κm)¯s assumes positive (negative) values towards the unburned (burned) gas side of the flame brush. For small values of Δ, the surface-weighted filtered value of curvature (κm)¯s approaches to κm (i.e., limΔ→0(κm)¯s=κm|∇c|/|∇c|=κm) and thus the ensemble averaged value of (κm)¯s remains small for small values of filter width as the ensemble averaged value of κm remains negligible for statistically planar flames. The difference between the ensemble averaged values of (κm)¯s and κm increases with increasing filter width Δ, as flame wrinkling increasingly takes place at the subgrid level. For the Le=1.0 flame (i.e., case D), the combination of positive (negative) value of (κm)¯s and weak (Sr+Sn)-κm and |∇c|-κm correlations gives rise to positive (negative) values of the ensemble averaged values of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 towards the unburned (burned) gas side of the flame brush for all values of Δ. The predominant positive (Sr+Sn)-κm and |∇c|-κm correlations give rise to positive values of the ensemble averaged values of ((Sr+Sn)∇.N→)¯sΣgen and Csg1 throughout the flame brush for small values of Δ in the Le=0.34, 0.6, and 0.8 flames. By contrast, negative (Sr+Sn)-κm and |∇c|-κm correlations (see Figures 4(a) and 4(b)) give rise to negative values of the ensemble averaged values of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 throughout the flame brush for small values of Δ in the Le=1.2 flame. These local dependences are progressively smeared with increasing Δ because of the convolution operation associated with LES filtering process, and this leads to positive (negative) values of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 towards the unburned (burned) gas side of the flame brush for all cases considered here, including the nonunity Lewis number flames where the curvature dependences of (Sr+Sn) and |∇c| are particularly strong.The dependences of(Sr+Sn)¯s and Σgen on 0.5×∂(Ni)¯s/∂xiΣgen are likely to capture some of κm dependences of (Sr+Sn) and |∇c| at small values of filter widths Δ (i.e., Δ<δth), where the flame is partially resolved. This effect is particularly prevalent in the nonunity Lewis number flames where both (Sr+Sn) and |∇c| are strongly correlated with curvature κm even though the flames are statistically planar in nature. As a result of this, the contribution of (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen remains close to that of ((Sr+Sn)∇·N→)¯sΣgen for small filter widths (i.e., Δ<δth) for the non-unity Lewis number flames, which is reflected in the small contribution of Csg1 (see Δ=0.8δth variations in Figures 3(a)–3(c) and Figure 3(e)). The correlation between the resolved quantities (e.g., dependences of (Sr+Sn)¯s and Σgen on 0.5×∂(Ni)¯s/∂xiΣgen) weakens with increasing filter width Δ due to smearing of local information. Moreover, physical processes take place increasingly at the subgrid level for Δ≫δth, and thus (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen does not capture the behaviour of ((Sr+Sn)∇·N→)¯sΣgen for large filter widths in all cases considered here, including the nonunity Lewis number flames where the curvature dependences of (Sr+Sn) and |∇c| are particularly strong. This leads to Csg1≈((Sr+Sn)∇·N→)¯sΣgen for Δ≫δth in all cases considered here (see Δ=2.4δth variations in Figures 3(f)–3(j)). It can be seen from Figure 3 that the positive contribution of Csg1 overcomes the negative contribution of Csg2 towards the unburned gas side of the flame brush in the Le=0.34 and 0.6 flames, which lead to positive value of Csg=Csg1+Csg2 towards the unburned gas side for all values of Δ in these cases (see Figure 2). By contrast, negative values of Csg2 overcome the positive contributions of Csg1 towards the unburned gas side of the flame brush in the Le=0.8, 1.0, and 1.2 flames, which lead to negative values of Csg=Csg1+Csg2 throughout the flame brush in these cases (see Figure 2).The subgrid fluctuations of the surface-weighted contributions of(Sr+Sn) and ∇·N→ are scaled here using SL and (Σgen-|∇c-|), respectively, to propose the following model for Csg1:
(14)Csg1=-β4(Σgen-|∇c-|)(c--c*)SLΣgen{exp⁡⁡[-aΣ(1-c-)]c-(1-c-)m},
where β4,c*,aΣ, and m are the model parameters. The function (c--c*)/{exp[-aΣ(1-c-))]c-(1-c-)m} in (14) is used to capture the correct qualitative behaviour of Csg1 throughout the flame brush. In a compressible, LES simulation c~ is readily available and c- needs to be extracted from c~.The methodology of extracting c- from c~ in the context of LES was discussed elsewhere [9, 10, 12] and will not be discussed in detail in this paper. The model parameter c* ensures that the transition from positive to negative value of Csg1 takes place at the correct location within the flame brush. The quantity (Σgen-|∇c-|) vanishes when the flow is fully resolved (i.e., limΔ→0(Σgen-|∇c-|)=limΔ→0(|∇c|¯-|∇c-|)=|∇c|-|∇c|=0.0), and thus Csg1 becomes exactly equal to zero when the flow is fully resolved (i.e., Δ→0) according to (14). It has been found that m=1.85 enables (14) to capture the qualitative behaviour of Csg1 when the optimum values of c*and aΣ are chosen. The optimum values of c*(aΣ) tend to increase with decreasing (increasing) Δ. The curvature κm dependences of (Sr+Sn) and |∇c| are influenced by Le (see Figures 4(a) and 4(b)), and these local dependences also appear in the resolved scale but their strength diminishes with increasing Δ due to filtering operation. As the resolved and subgrid curvature terms are closely related [9, 10, 12], the qualitative behaviour of Csg1 is also affected by the curvature dependences of displacement speed components and scalar gradient at the resolved scale, which leads to the variation of the optimum values of aΣ, β4, and c* with Le and Δ. The model parameter β4 needs to be deceased for decreasing values of Σgen for satisfactory prediction of (14). The prediction of (14) ensemble averaged on c~ isosurfaces is compared with the ensemble averaged values of Csg1 in Figure 5 for all cases considered here for the optimum values of β4,c*, and aΣ for Δ=0.8δth and Δ=2.4δth when m is taken to be m=1.85. The optimum values of β4,c*, and aΣ are estimated by calibrating the prediction of (14) with respect to the ensemble averaged values of Csg1 obtained from DNS data and the variation of the optimum values of β4/Σgen, c*, and aΣ with Δ/δth for all cases are shown in Figure 6. The optimum values of β4/Σgen, c*, and aΣ are parameterised here as
(15a)β4Σgen=9.81δth[l1+(l2-l1){1.0+exp⁡⁡[-10.0(Le-1)]}1/2],
where
(15b)l1=1.2[Δ2.79+1.2(Δ+δth)2.79](Δ+δth)2.79;l2=1.34[Δ0.67+0.53(Δ+δth)0.67][3.1Δ0.67+0.1(Δ+δth)0.67],(15c)c*=k1+[(k2-k1){1.0+exp⁡⁡(-2.0(Δ/δth-1.5))}];aΣ=k4(1.0+exp⁡⁡(-5.0(Δ/δth-1.0))),
where
(15d)k1=0.75+0.15[1.0+exp⁡⁡(-5.0(k3-4.6))];k2=0.65+0.05[1.0+exp⁡⁡(-9.0(k3-4.0))],(15e)k4=0.81-0.67[1.0+exp⁡⁡(-5.0(k3-4.6))];k3=(ReΔ0.83+0.1)[(Δ/δth)1.73+0.1];ReΔ=4ρ0Δμ0(2k~3).Variations ofCsg1 (solid line) and Csg2 (dashed line) conditionally averaged in bins of c~ across the flame brush along with the predictions of (14) (vertical line) and (16) (crosses line) for Δ=8Δm≈0.8δth (top row): (a) Le=0.34 (1st column); (b) Le=0.6 (2nd column); (c) Le=0.8 (3rd column); (d) Le=1.0 (4th column); (e) Le=1.2 (5th column) and for filter size Δ=24Δm≈2.4δth (bottom row): (f) Le=0.34 (1st column); (g) Le=0.6 (2nd column); (h) Le=0.8 (3rd column); (i) Le=1.0 (4th column); (j) Le=1.2 (5th column).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)Variations of the model parametersβ1 (◯), β2(□), β3(∆), β4/Σgen (down-pointing triangle), β5(+), aΣ(*), and c*(×) with Δ for: (a) Le=0.34; (b) Le=0.6; (c) Le=0.8; (d) Le=1.0; (e) Le=1.2.
(a)
(b)
(c)
(d)
(e)Figure5 shows that (14) satisfactorily predicts Csg1 when m is taken to be m=1.85, and the optimum values of β4,c*, and aΣ are used. According to the parameterisation given by (15a)–(15e), β4 increases with decreasing Le, as the effects of chemical reaction strengthen with decreasing values of Lewis number (see Table 2). Moreover, β4/Σgen, c*, and aΣ approach to asymptotic values for large values of Δ and turbulent Reynolds number based on LES filter width ReΔ=4ρ02k~/3Δ/μ0, where ρ0 and μ0 are the unburned gas density and viscosity, respectively.Here, the contribution of(Dκm2)¯s-(D∂Ni/∂xi)¯s∂(Ni)¯s/∂xi is scaled with (ΞΔ-1)nSLΣgen (i.e., (Dκm2)¯s-(D∂Ni/∂xi)¯s∂(Ni)¯s/∂xi~(ΞΔ-1)nSLΣgen) where the sub-grid fluctuations of D are taken to scale with SL/Σgen. The above relations are utilised here to propose a model for Csg2 in the following manner:
(16)Csg2=-β5SL(ΞΔ-1)nΣgen2c-(1-c-),
where ΞΔ=Σgen/|∇c-| is the wrinkling factor [8, 11, 43, 50, 51], β5 and n are the model parameters, and c-(1-c-) is used to capture the correct qualitative behaviour of Csg2. The subgrid curvature term Csg2 vanishes when the flow is fully resolved according to (16), (i.e., limΔ→0Ξ=limΔ→0Σgen/|∇c-|=limΔ→0|∇c|¯/|∇c-|=|∇c|/|∇c|=1.0). It has been found that (16) satisfactorily captures the behaviour of Csg2 throughout the flame brush for n=1.0 in all cases when a suitable value of β5 is used. The variation of the global mean optimum values of β5 with Δ/δth is shown in Figure 6 for all cases considered here. The optimum values of β5 have been parameterised here in the following manner:
(17a)β5=m(Le){ReΔ(ReΔ+1.0)}×[r1+{(r2-r1)(1.0+exp⁡⁡(-5.0(ReΔ-r3)))}],
where
(17b)r1=1.6(r41.23+6.24)(7.17r41.23+0.26);r2=1.88(r42.27+5.92)(8.47r42.27+0.47)(17c)r3=35.0erf⁡⁡[exp⁡⁡{5.3(r4-1.0)}];r4=Δ(Δ+δth)(17d)m(Le)=(r5+(1.0-r5){1.0+exp⁡⁡[-10.0(Le-1.0)]}1/4);r5=0.46[r45.22+4.53][8.0r45.22+2.96].The predictions of(16) ensemble averaged on c~ isosurfaces are compared with ensemble averaged values of Csg2 in Figure 5 for all cases at Δ=0.8δth and Δ=2.4δth, which show that (16) satisfactorily predicts the statistical behaviour of Csg2 when n is taken to be n=1.0 and the optimum value of β5 is used. According to (17a)–(17d), β5 approaches to asymptotic values for large values of Δ and ReΔ=4ρ02k~/3Δ/μ0.Equations(13) and (15a)–(15e) can be combined to propose a model for Csg in the following manner:
(18)Csg=-β4(Σgen-|∇c-|)(c--c*)SLΣgen{exp⁡⁡[-aΣ(1-c-)]c-(1-c-)m}-β5SL(ΞΔ-1)nΣgen2c-(1-c-).The above model will henceforth be referred to CSGNEW model in this paper. Equation(18) allows for a positive contribution of Csg through the contribution of -β4(Σgen-|∇c-|)(c--c*)SLΣgen/{exp[-aΣ(1-c-)]c-(1-c-)m}, which is absent in the CSGCAND, CSGCANT, and CSGCHAR models. The predictions of the CSGCAND, CSGCANT, CSGCHAR, and CSGNEW models for Δ=0.8δth and Δ=2.4δth are compared with Csg obtained from DNS in Figure 7 for the optimum values of β1, β2, β3, and β5. The optimum values of β1, β2, and β3 are estimated by calibrating the models based on the ensemble averaged value of Csg obtained from DNS data. The variations of the optimum values of β1, β2, and β3 with Δ for all cases are also shown in Figure 6. It is evident from Figure 6 that β1, β2, β3, and β5 remain greater than unity for all cases. This is found to be consistent with the realisability analysis by Hawkes and Cant [52]. Figure 6 further demonstrates that the optimum values of β1, β2, and β3 change appreciably with increasing Δ, which is consistent with earlier findings [9, 10, 12]. Moreover, optimum values of β1, β2, and β3 for a given Δ are affected by Le (see Figure 6). It is worth noting that parameterisation of the optimum values of β1, β2 and β3 also yields complex relations similar to (15a)–(15e) and (17a)–(17d). However, such parameterisation is not presented here because the CSGCAND, CSGCANT, and CSGCHAR models do not capture the qualitative behaviour of Csg for the Le=0.34 and 0.6 flames.Variation ofCsg (solid line) conditionally averaged in bins of c~ across the flame brush along with the predictions of CSGCAND (asterisk line), CSGCANT (dashed line), CSGCHAR (squared line) and CSGNEW (triangle line) for filter sizes Δ=8Δm≈0.8δth (top row): (a) Le=0.34 (1st column); (b) Le=0.6 (2nd column); (c) Le=0.8 (3rd column); (d) Le=1.0 (4th column); (e) Le=1.2 (5th column) and for filter size Δ=24Δm≈2.4δth (bottom row): (f) Le=0.34 (1st column); (g) Le=0.6 (2nd column); (h) Le=0.8 (3rd column); (i) Le=1.0 (4th column); (j) Le=1.2 (5th column).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)It can further be seen from Figure7 that the CSGCHAR model tends to overpredict the negative values of Csg towards the unburned gas side in cases C–E (Le = 0.8, 1.0 and 1.2 flames), and this behaviour becomes more prominent with increasing filter size. It is clear from Figure 7 that for Δ=24Δm=2.4δth, the CSGCHAR model predicts the maximum magnitude of Csg near the middle of the flame whereas the actual maximum magnitude of Csg is attained slightly towards the burned gas side. The CSGCAND and CSGCANT models give comparable performance for optimum values of β1 and β2 in cases C–E. However, the CSGCAND and CSGCANT models do not satisfactorily capture the qualitative behaviour of Csg and underpredict (overpredict) the magnitude of Csg towards the burned gas side (middle) of the flame brush in the Le = 0.8, 1.0 and 1.2 flames. Figure 7 demonstrates that the CSGNEW model captures the qualitative behaviour of Csg in a better manner than the CSGCAND and CSGCANT models and the quantitative agreement between Csg and the CSGNEW model remains better than the CSGCAND, CSGCANT, and CSGCHAR models in all cases for all values of Δ when optimum values of β4, β5, aΣ, and c* are used.


## 5. Conclusions

The LES modelling of the curvature term(Sd∇·N→)¯sΣgen of the generalised FSD Σgen transport equation has been addressed here using a simplified chemistry-based DNS database of freely propagating statistically planar turbulent premixed flames with Lewis number Le ranging from 0.34 to 1.2. The statistical behaviours of the subgrid curvature term Csg for a range of different values of Δ have been analysed in terms of its contributions Csg1 and Csg2 arising from the combined reaction and normal diffusion component and tangential diffusion components of displacement speed (i.e., (Sr+Sn) and St=-2Dκm), respectively. The Lewis number is shown to have significant influences on the statistical behaviours of the resolved and subgrid components of the FSD curvature term. Detailed physical explanations have been provided for the observed filter size and Lewis number dependences of the different components of (Sd∇·N→)¯sΣgen. Models have been identified for individual components of the subgrid curvature term (i.e., Csg1 and Csg2), and the performances of these models have been compared to the corresponding quantities extracted from DNS data. It has been found that the new models for Csg1 and Csg2 satisfactorily capture the statistical behaviours of the corresponding terms extracted from DNS data. It has been found that the existing models for the subgrid curvature term Csg do not satisfactorily capture the qualitative behaviour of the corresponding quantity extracted from DNS data for all the flames considered here. This problem is particularly prevalent for flames with small values of Lewis number (e.g., Le=0.34 and 0.6) where Csg locally assumes positive values, whereas the existing models can only predict negative values of Csg. The performance of the newly proposed model for Csg has been found to be better than the existing models, and it has been shown to capture positive contributions of Csg for the Le≪1 flames. The present analysis has been carried out using a DNS database with moderate value of Ret in the absence of the effects of detailed chemistry and transport. As simplified chemistry-based DNS qualitatively captures the curvature κm=∇·N→/2 and strain rate dependences of Sd and |∇c| obtained from detailed chemistry based simulations, it can be expected that the statistical behaviours of the curvature term (Sd∇·N→)¯sΣgen presented in this paper will be valid at least in a qualitative sense in the context of detailed chemistry. However, the quantitative values of the model parameters (i.e., β4, β5, aΣ, and c*) may need to be altered in the presence of detailed chemistry. Thus, three-dimensional DNS data with detailed chemistry and experimental data at higher values of Ret will be necessary for more comprehensive modelling of (Sd∇·N→)¯sΣgen and Csg in the context of LES. Moreover, the newly proposed models need to be implemented in LES simulations for the purpose of a posteriori assessments. However, this is kept beyond the scope of this paper. Several previous studies [3–7, 9–12, 43–49] concentrated purely on the model development based on a priori analysis of DNS data and the same approach has been adopted here. Implementation of the newly developed models in LES simulations will form the basis of future investigations.



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*Source: 103727-2012-08-15.xml*