A simplified chemistry based three-dimensional Direct Numerical Simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different values of turbulent Reynolds number has been used for the a priori modelling of the curvature term of the generalised Flame Surface Density (FSD) transport equation in the context of Large Eddy Simulation (LES). The curvature term has been split into the contributions arising due to the reaction and normal diffusion components of displacement speed and the term originating from the tangential diffusion component of displacement speed. Subsequently, these contributions of the curvature term have been split into the resolved and subgrid contributions. New models have been proposed for the subgrid curvature terms arising from the combined reaction and normal diffusion components and the tangential diffusion component of displacement speed. The performances of the new model and the existing models for the subgrid curvature term have been compared with the corresponding quantity extracted from the explicitly filtered DNS data. The new model for the subgrid curvature term is shown to perform satisfactorily in all cases considered in the current study, accounting for wide variations in LES filter size.
1. Introduction
Flame Surface Density (FSD) based reaction rate closure is one of the popular methods of turbulent premixed combustion modelling in the context of Reynolds Averaged Navier Stokes (RANSs) simulations [1, 2]. The FSD based modelling has recently been extended to Large Eddy Simulations (LESs) [3–12]. The generalised FSD Σgen is defined as Σgen=|∇c|¯ [3–10] where c is the reaction progress variable and the overbar indicates a LES filtering operation. The transport equation of Σgen is given by [1, 4–7, 9, 11]:
(1)∂Σgen∂t+∂(u~jΣgen)∂xj=-∂[((ui)¯s-u~i)Σgen]∂xi+((δij-NiNj)∂ui∂xj)¯sΣgen-∂[(SdNi)¯sΣgen]∂xi+(Sd∂Ni∂xi)¯sΣgen,
where Ni=-(∂c/∂xi)/|∇c| is the ith component of flame normal vector and Sd=(Dc/Dt)/|∇c| is the displacement speed, (Q)¯s=Q|∇c|¯/Σgen and Q~=ρQ¯/ρ- are the surface-weighted and Favre filtered values of a general quantity Q. The final term on the right hand side of (1) originates due to flame curvature κm=(∂Ni/∂xi)/2 and thus this term (i.e., (Sd∇·N→)¯sΣgen) is referred to as the curvature term [4–7, 9, 11]. It is evident from (1) that the curvature dependence of Sd plays a key role in the statistical behaviours of (Sd∇·N→)¯sΣgen and this was confirmed in previous a priori Direct Numerical Simulation (DNS) based analyses [9, 11]. It was previously demonstrated [9, 11] that the existing models for the subgrid curvature term Csg often do not capture its correct qualitative and quantitative behaviours, particularly in the Thin Reaction Zones (TRZ) regime flames. Moreover, the model parameters for the existing Csg models were found to be strong functions of LES filter width Δ [9, 11]. The modelling of (Sd∇·N→)¯sΣgen therefore remains one of the weakest points in the LES modelling of the Σgen transport equation. This gap in the existing literature is addressed in this paper by explicitly LES filtering a DNS database of freely propagating statistically planar turbulent premixed flames with different values of turbulent Reynolds number Ret. In this regard, the main objectives of the present study are as follows:
to analyse the statistical behaviours of the subgrid FSD curvature term in the context of LES for the flames with different values of Ret;
to propose models for the subgrid FSD curvature term and assess their performances in comparison to the corresponding quantities extracted from DNS data.
The necessary mathematical background and numerical details will be provided in the next section. Following this, the results will be presented and subsequently discussed. Finally the main findings will be summarised and conclusions will be drawn.
2. Mathematical Background and Numerical Implementation
Although three-dimensional DNS with detailed chemical mechanism is currently possible, it remains extremely computationally intensive [13] and is often not suitable for a detailed parametric analysis. Thus the chemical mechanism is simplified here using a single step Arrhenius type chemical reaction in order to carry out a parametric variation in terms of Ret. For the convenience of modelling, (Sd∇·N→)¯sΣgen is often split as [4–7, 9, 11]
(2)(Sd∇·N→)¯sΣgen=Cmean+Csg,
where Cmean and Csg are the resolved and subgrid curvature terms, respectively. Chakraborty and Cant [9, 11] analysed the possibility of using three different expressions of Cmean:
(3)Cmean=(Sd)¯s[∂(Ni)¯s∂xi]Σgen,Cmean=(Sd)¯s[∂Mi∂xi]Σgen,Cmean=(δij-(NiNj)-s)∂[(Sd)¯s(Ni)¯s]∂xjΣgen,
where (Ni)¯s=-(∂c-/∂xi)/Σgen and Mi=-(∂c-/∂xi)/|∇c-| are the ith component of surface-weighted and resolved flame normal vector, respectively. Previous a priori DNS analyses suggested that Cmean=(Sd)¯s[∂(Ni)¯s/∂xi]Σgen is the most preferred expression for the resolved curvature term out of the three options presented in (3), as it allows for the smallest magnitude of Csg, while satisfactorily capturing the qualitative behaviour of (Sd∇·N→)¯sΣgen [9, 11]. Moreover, Cmean=(Sd)¯s[∂(Ni)¯s/∂xi]Σgen was used in previous LES simulations [5–7, 12]. It is useful to split Sd=(Dc/Dt)/|∇c|=[w˙+∇·(ρD∇c)]/ρ|∇c| in the following manner [9–11, 14, 15] for obtaining further insight into (Sd∇·N→)¯sΣgen:
(4)Sr=w˙ρ|∇c|,Sn=N→·∇(ρDN→.∇c)ρ|∇c|,St=-D∇·N→=-2Dκm,
where w˙ is the reaction rate and D is the progress variable diffusivity. The following expression for Csg can be obtained using (4) and Cmean=(Sd)¯s[∂(Ni)¯s/∂xi]Σgen:
(5)Csg=Csg1+Csg2=(Sd∂Ni∂xi)¯sΣgen-(Sd)¯s∂(Ni)¯s∂xiΣgen,
where
(6i)Csg1=((Sr+Sn)∂Ni∂xi)¯sΣgen-(Sr+Sn)¯s∂(Ni)¯s∂xiΣgen,(6ii)Csg2=[-(D(∂Ni∂xi)2)¯sΣgen-(D∂Ni∂xi)¯s∂(Ni)¯s∂xiΣgen].
Equation (6i) indicates that the curvature κm=∇·N→/2 dependences of (Sr+Sn) and |∇c| are likely to influence the statistical behavior of Csg1. According to (6ii), Csg2 remains deterministically negative throughout the flame brush. Hawkes and Cant [6, 7] modified a version of the coherent flamelet model by Candel et al. [2] as
(7)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 the RANS model proposed by Cant et al. [1] for the purpose of LES as:
(8)Csg=-CHSLΣgen2(1-c-),
where CH=αNβ2(1-(1/3)[1-exp(-10(1-c-)k~/ΣgenSLΔ)]) and k~=(ρuiui¯-ρ-u~iu~i)/2ρ- is the subgrid kinetic energy and β2 is a model parameter. Charlette et al. [4] modelled Csg as
(9)Csg=-β3SL(Σgen-|∇c-|)Σgenc-(1-c-),
where β3 is a model parameter. The models given by (7)–(9) (henceforth will be referred to as CSGCFM, CSGCPB, and CSGCHAR, respectively) ensure that Csg vanishes when the flow is fully resolved (i.e., (Nk)¯s(Nk)¯s=1.0 and Σgen=|∇c-|). Modelling of Csg1 and Csg2 using a priori analysis of DNS data, and the assessment of the models given by (7)–(9), will be addressed in Section 3 of this paper.
In the present study a compressible DNS database of freely propagating statistically planar turbulent premixed flames under decaying turbulence has been considered. The simulation domain of size 36.6δth×24.1δth×24.1δth, was discretised using a Cartesian mesh of size 345×230×230 with uniform mesh spacing in each direction where δth=(Tad-T0)/Max|∇T^|L is the thermal flame thickness with Tad, T0, and T^ being the adiabatic flame, unburned gas, and instantaneous gas temperatures, respectively, and the subscript L refers to the unstrained planar laminar flame quantities. The domain boundaries in the direction of mean flame propagation (i.e., x1-direction) are taken to be partially nonreflecting, whereas the transverse boundaries are taken to be periodic. The partially nonreflecting boundaries are specified using the well-known Navier Stokes Characteristic Boundary Conditions (NSCBC) technique [16]. The simulations have been carried out using a three-dimensional compressible DNS code called SENGA [17] A 10th order central difference scheme is used for spatial differentiation for internal grid points, and the order of differentiation decreases gradually to a one-sided 2nd order scheme towards nonperiodic boundaries [17]. A third order Runge-Kutta scheme was used for the purpose of time advancement [17]. For all cases, the reacting flow field is initialised by a steady unstrained planar laminar flame solution, and the initial turbulent velocity fluctuations are specified using an initially homogeneous isotropic velocity field. About 10 grid points are kept within the thermal flame thickness δth for all cases considered here. The initial values for the root-mean-square turbulent velocity fluctuation normalised by unstrained planar laminar burning velocity u′/SL and the integral length scale to flame thickness ratio l/δth are presented in Table 1 along with the values of Damköhler number Da=l.SL/u′δth, Karlovitz number Ka=(u′/SL)3/2(l/δth)-1/2, and turbulent Reynolds number Ret=ρ0u′l/μ0, where ρ0 and μ0 are the unburned gas density and viscosity, respectively. The heat release parameter τ=(Tad-T0)/T0 and Lewis number Le are taken to be 4.5 and 1.0 for all cases considered here. Standard values are taken for Prandtl number Pr, ratio of specific heats γ, and the Zel’dovich number β=Tac(Tad-T0)/Tad2 (i.e., Pr=0.7, γ=1.4, and β=6.0), where Tac is the activation temperature. The turbulent Reynolds number Ret scales as Ret~Da2Ka2, and thus the variation of Ret in cases A–E is brought about by modifying Da and Ka independently from each other. In cases A, C and E, Da is held constant, while Ka is held constant in cases B, C, and D. For all cases the Karlovitz number remains greater than unity indicating the TRZ regime combustion according to the regime diagram by Peters [18]. The range of Ret values considered in this study remains modest, although several previous studies [3, 9, 11, 15, 19–23] with comparable values of Ret have made valuable contributions to the fundamental understanding and the modelling of turbulent premixed combustion. Moreover, the range of Ret considered here is comparable to that of previous laboratory-scale experiments [24].
List of initial simulation parameters and non-dimensional numbers.
Case
u'/SL
l/δth
Ret
Da
Ka
A
5.0
1.67
22
0.33
6.54
B
6.25
1.44
23.5
0.23
9.84
C
7.5
2.5
49.0
0.33
9.82
D
9.0
4.31
100.0
0.48
9.83
E
11.25
3.75
110
0.33
14.73
In all cases flame-turbulence interaction takes place under decaying turbulence. The simulations were run for a time equal to one chemical time scale (i.e., tc=δth/SL), which is equivalent to 2.0tf in case D; 3.0tf in cases A, C, and E; 4.34tf for case B. The aforementioned simulation times remain comparable to several studies [3, 9, 11, 15, 19–23], which contributed to the FSD based modelling in the past. The global turbulent kinetic energy and burning rate were not varying significantly with time when statistics were extracted (see Figure 1 of [23]) and the qualitative nature of the statistics was found to have remained unchanged since t=1.0l/u′ for all cases [23]. At time t=δth/SL, the global level of u′/sl had decayed from the initial values by about 45%, 55%, 40%, 25%, and 32% in cases A–E, respectively. The values of l/δth had increased from their initial values by a factor of about 1.5–2.25 at t=δth/SL, but there were still enough turbulent eddies on each side of the computational domain [23]. Values for u′/SL and l/δth at the time when statistics were extracted were presented in Table 2 of [23] and are not repeated here. The flame thickness δth remained greater than the Kolmogorov length scale η for all cases when the statistics were extracted (see Table 2 of [23]), confirming the TRZ regime combustion.
For the purpose of a priori DNS analysis, the relevant quantities are explicitly filtered using a Gaussian kernel G(r→)=(6/πΔ2)3/2exp(-6r→·r→/Δ2). The filtered value of a general quantity Q-(x→,t) is evaluated using the following convolution operation: Q-(x→,t)=∫Q(x→-r→)G(r→)dr→. The statistical behaviours of the FSD curvature term (Sd∇·N→)¯sΣgen have been analysed here for Δ ranging from Δ=4Δm≈0.4δth to Δ=24Δm≈2.4δth, where Δm is the DNS mesh size (Δm≈0.1δth). These filter sizes are comparable to the range of Δ used in a priori DNS analysis in several previous studies [3, 4, 9–11] and span a useful range of length scales (i.e., from Δ comparable to 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).
3. Results and Discussion
The isosurfaces of c ranging from 0.01 to 0.99 at time t=δth/SL for all cases are shown in Figure 1. A comparison between Figures 1(a)–1(e) reveals that the wrinkling of the flame surface with increasing u′/SL~Ret1/4Ka1/2~Ret1/2/Da1/2. Figures 1(a)–1(e) further demonstrate that the flame surfaces in all cases show a range of different curvatures and this range increases with increasing u′/SL~Ret1/4Ka1/2~Ret1/2/Da1/2. This indicates that the interrelations between Sd and κm and between |∇c| and κm may lead to nonnegligible value of (Sd∇·N→)¯sΣgen even for statistically planar flames.
Instantaneous isosurfaces of c ranging from 0.01 to 0.99 at t=1.0δth/SL for cases (a)–(e) A–E.
The variations of the ensemble averaged values of (Sd∇·N→)¯sΣgen conditional on c~ isosurfaces for all cases are shown in Figure 2 for Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth. The filter widths Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth correspond to two representative situations, where the flame is partially resolved and where the flame is fully unresolved, respectively. Figure 2 shows that (Sd∇·N→)¯sΣgen assumes predominantly negative values throughout the flame brush. Although the resolved curvature term Cmean=(Sd)¯s∂(Ni)¯s/∂xiΣgen captures the qualitative behaviour of (Sd∇·N→)¯sΣgen throughout the flame brush, the magnitude of Cmean remains smaller than the magnitude of (Sd∇·N→)¯sΣgen for the major portion of the flame brush in all cases for all values of Δ (see Figure 2). This leads to predominantly negative values of Csg, although the ensemble averaged values of Csg on c~ isosurfaces exhibits positive values towards the unburned gas side of the flame brush for the flames with low and moderate values of turbulent Reynolds number (e.g., cases A–C). By contrast, the variation of ensemble averaged values of Csg on c~ isosurfaces exhibits only negative values throughout the flame brush for the flames with high values of turbulent Reynolds number (e.g., cases D and E). The models of Csg given by (7)–(9) only predicts negative values of Csg and thus will not be capable of predicting the positive values of Csg in cases A–C.
Variations of (Sd∇·N→)¯sΣgen, Cmean=(Sd)¯s∂(Ni)¯s/∂xiΣgen and Csg conditionally averaged in bins of c~ across the flame brush for Δ=8Δm≈0.8δth (top row) and Δ=24Δm≈2.4δth (bottom row) for cases A (1st column), B (2nd column), C (3rd column), D (4th column), and E (5th column). All the curvature terms in this and subsequent figures are normalised by SL/δth2.
Comparing (Sd∇·N→)¯sΣgen, Cmean, and Csg magnitudes for Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth reveals that the magnitudes of (Sd∇·N→)¯sΣgen and Cmean decrease with increasing Δ, whereas the relative magnitude of Csg in comparison to Cmean increases with increasing Δ. This observation is consistent with previous findings [9, 11]. The smearing of local information as a result of the weighted-averaging process involved in LES filtering leads to the decrease in the magnitudes of (Sd∇·N→)¯sΣgen and Cmean for increasing values of Δ. The flow becomes increasingly unresolved with increasing Δ and thus the flame curvature and its influence on the FSD evolution are increasingly felt at the subgrid scale, which is reflected in the relatively high magnitudes of Csg in comparison to Cmean for large values of Δ.
It is useful to examine the statistical behaviours of Csg1 and Csg2 in order to explain the differences in the behaviours of Csg for flames with different Ret. The variations of the ensemble averaged values of ((Sr+Sn)∇·N→)¯sΣgen, (-(D(∇·N→)2)¯sΣgen=-4(Dκm2)¯sΣgen), Csg1 and Csg2 conditional on c~ isosurfaces are shown in Figure 3 for cases A–E for Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth, respectively. Figure 3 demonstrates that both ((Sr+Sn)∇·N→)¯sΣgen and Csg1 remain predominantly positive (negative) towards the unburned (burned) gas side of the flame brush for all values of Δ considered here. The contribution of -4(Dκm2)¯sΣgen and Csg2 remains deterministically negative throughout the flame brush (see Figure 3). It is evident from Figure 3 that -4(Dκm2)¯sΣgen remains a leading order contributor to (Sd∇·N→)¯sΣgen for all the flames at all values of Δ (see Figure 3), which is consistent with the expected behaviour in the TRZ regime, where -4(Dκm2)¯sΣgen is expected to play an important role [18]. Figure 3 further shows that Csg1 remains close to the magnitude of ((Sr+Sn)∇·N→)¯sΣgen for all Δ for all cases considered here, indicating that (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen does not play a major role in capturing the behaviour of ((Sr+Sn)∇·N→)¯sΣgen. By contrast, 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, Csg2 remains the leading order contributor to -4(Dκm2)¯sΣgen, indicating that (-(D∂Ni/∂xi)¯s∂(Ni)¯s/∂xiΣgen) plays a 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 when the flame is partially resolved. Figure 3 further shows that the order of magnitudes of both Csg1 and Csg2 remains comparable for large values of Δ (i.e., Δ>δth) and thus accurate modelling of Csg1 and Csg2 is necessary for accurate modelling of Csg. As the range of κm values obtained on a flame surface increases with increasing flame wrinkling at higher values of u′/SL~Ret1/4Ka1/2~Ret1/2/Da1/2, the magnitude of -4(Dκm2)¯s increases with increasing Ret, which in turn leads to increasing magnitude of -4(Dκm2)¯sΣgen and Csg2 with increasing Ret for a given value of Da or Ka (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 flames with small and moderate values of turbulent Reynolds number (i.e., cases A–C) and yields a net positive contribution of Csg towards the reactant side of the flame brush (see Figure 2).
Variations of ((Sr+Sn)∇·N→)¯sΣgen, -4(Dκm2)¯sΣgen, Csg1, and Csg2 conditionally averaged in bins of c~ across the flame brush for Δ=8Δm≈0.8δth (top row) and Δ=24Δm≈2.4δth (bottom row) for cases A (1st column), B (2nd column), C (3rd column), D (4th column), and E (5th column).
The statistical behaviours of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 depend on the nature of the correlations between (Sr+Sn) and κm=∇·N→/2 and between |∇c|, κm, and the variation of (κm)¯s across the flame brush. The correlation coefficients for the (Sr+Sn)-κm and |∇c|-κm dependences for five different c isosurfaces across the flame brush for all cases are shown in Figures 4(a) and 4(b), respectively. For unity Lewis number flames St=-2Dκm is deterministically negatively correlated with κm with a correlation coefficient equal to −1.0. Figure 4(a) suggests that (Sr+Sn)-κm correlation is much weaker than the St-κm correlation in all cases. Moreover, Figure 4(b) demonstrates that |∇c| and κm remain weakly correlated throughout the flame brush for all cases. Figures 4(a) and 4(b) demonstrate that the curvature dependences of (Sr+Sn) and |∇c| remain qualitatively similar for all the flames [25]. The physical explanations of the observed curvature dependences of (Sr+Sn) and |∇c| have been discussed elsewhere [25] and will not be repeated here.
Correlation coefficients for the (a) (Sr+Sn)-κm and (b) |∇c|-κm correlations for c=0.1,0.3,0.5,0.7 and 0.9 isosurfaces for cases A–E. Variations of (κm)¯s×δth conditionally averaged in bins of c~ across the flame brush for (c) Δ=8Δm≈0.8δth and (d) Δ=24Δm≈2.4δth.
The variation of the ensemble averaged values of (κm)¯s conditional on c~ isosurfaces for all cases are shown in Figures 4(c) and 4(d) for Δ=8Δm≈0.8δth and Δ=24Δm≈2.4δth, respectively, which demonstrates that (κm)¯s predominantly assumes positive (negative) values towards the unburned (burned) gas side of the flame brush and the magnitude of (κm)¯s increases with increasing Δ. The quantity (κm)¯s approaches to κm for small values of Δ (i.e., limΔ→0(κm)¯s=limΔ→0κm|∇c|¯/|∇c|¯=κm|∇c|/|∇c|=κm) and the mean value of κm=∇·N→/2 remains negligible for all the c isosurfaces due to the statistical planar nature of the flames. However, subgrid level curvature increases with increasing Δ and thus the magnitude of (κm)¯s increases with increasing values of Δ. Relatively weak curvature dependences of (Sr+Sn) and |∇c| lead to positive (negative) values of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 towards the unburned (burned) gas side of the flame brush due to positive (negative) value of (κm)¯s. The contribution of resolved curvature term (Sr+Sn)¯s∂(Ni)¯s/∂xiΣgen remains negligible in comparison to ((Sr+Sn)∇·N→)¯sΣgen, due to relatively small values of ∂(Ni)¯s/∂xi in comparison to 2(κm)¯s=(∂Ni/∂xi)¯s in statistically planar flames. Thus the contributions of ((Sr+Sn)∇·N→)¯sΣgen and Csg1 remain close to each other for all values of Δ (see Figure 3).
The subgrid fluctuations of the surface-weighted contributions of (Sr+Sn) and ∇·N→ are taken to scale with SL and (Σgen-|∇c-|), respectively, to propose the following model for Csg1 in this analysis:
(10)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 (10) is used to capture the correct qualitative behaviour of Csg1 across 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, 11] 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 (10). It has been found that m=1.85 enables (10) to capture the qualitative behaviour of Csg1 when the optimum values of c* and aΣ are chosen. The optimum value of c*(aΣ) tends to increase with decreasing (increasing) Δ. The κm dependences of (Sr+Sn) and |∇c| are reflected mostly in the resolved scale but these effects weaken with increasing values of Δ [9, 11]. As the resolved and subgrid curvature terms are closely related [9, 11], the qualitative behaviour of Csg1 is also affected by the κm dependences of (Sr+Sn) and |∇c|, which leads to the variation of the optimum values of aΣ and c*. The model parameter β4 is found to decrease with decreasing values of Σgen for satisfactory quantitative prediction of Csg1, which is accounted for by expressing β4 as β4=9.80Σgen×δth. The prediction of (10) ensemble averaged on c~ isosurfaces is compared with the ensemble averaged values of Csg1 in Figure 5 for all cases for the optimum values of c* and aΣ, for Δ=0.8δth and Δ=2.4δth, when β4 and m are taken to be β4=9.80Σgen×δth and m=1.85. The optimum values of c* and aΣ are estimated by calibrating the prediction of (10) with respect to the values of Csg1 obtained from DNS data and the variation of the global mean optimum values of c* and aΣ with Δ/δth for all cases are shown in Figure 6. The optimum values of c* and aΣ are parameterised here as
(11i)c*=k1+[(k2-k1){1.0+exp(-2.0(Δ/δth-1.5))}],aΣ=k3(1.0+exp(-5.0(Δδth-1.0))),
where
(11ii)k1=0.75+0.15[1.0+exp(-5.0(k4-4.6))],k2=0.65+0.05[1.0+exp(-9.0(k4-4.0))],(11iii)k3=0.81-0.67[1.0+exp(-5.0(k4-4.6))],k4=(ReΔ0.83+0.1)[(Δ/δth)1.73+0.1],whereReΔ=4ρ02k~3Δμ0.
Figure 5 shows that (10) satisfactorily predicts Csg1 when β4=9.80Σgen×δth and m=1.85, and (11i), (11ii), and (11iii) are used for c* and aΣ.
Variations of Csg1 and Csg2 conditionally averaged in bins of c~ across the flame brush for Δ=8Δm≈0.8δth (top row) and Δ=24Δm≈2.4δth (bottom row) for cases A (1st column), B (2nd column), C (3rd column), D (4th column), and E (5th column) along with the predictions of (10) and (12).
Variations of the model parameters β1, β2, β3, β5, aΣ and c* with Δ/δth for cases: (a–e) A–E.
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~[(Σgen/|∇c-|)-1]nSLΣgen), where the subgrid fluctuations of D is taken to scale with SL/Σgen (i.e., D~SL/Σgen). The above relations are utilised here to propose a model for Csg2 (see (6ii)) in the following manner:
(12)Csg2=-β5SL(ΞΔ-1)nΣgen2c-(1-c-),
where ΞΔ=Σgen/|∇c-| is the wrinkling factor [8, 10, 19], β5 is a model parameter, and c-(1-c-) is used to capture the correct qualitative behaviour of Csg2. According to (12), Csg2 vanishes when the flow is fully resolved (i.e., limΔ→0Ξ=limΔ→0Σgen/|∇c-|=limΔ→0|∇c|¯/|∇c-|=|∇c|/|∇c|=1.0). It has been found that (12) satisfactorily captures the behaviour of Csg2 throughout the flame brush for n=1.0 in all cases considered here when a suitable value of β5 is used. The variation of the global mean optimum values of β5 with Δ/δth for all cases is shown in Figure 6. The optimum values of β5 has been parameterised here in the following manner:
(13i)β5={ReΔ(ReΔ+1.0)}×[r1+{(r2-r1)1.0+exp(-5.0(ReΔ-r3))}],
where
(13ii)r1=1.6(r41.23+6.24)(7.17r41.23+0.26),r2=1.35(r43.53+6.10)(13.25r43.53+0.56),(13iii)r3=35.0erf[exp{5.3×(1.37r4-1.0)}],r4=Δ(Δ+δth).
The predictions of (12) ensemble averaged on c~ isosurfaces are compared with the ensemble averaged values of Csg2 in Figure 5 all for cases for Δ=0.8δth and Δ=2.4δth, which show that (12) satisfactorily predicts the statistical behaviour of Csg2 when n=1.0 and (13i), (13ii), and (13iii) are used for β5.
Equations (10) and (12) can be combined to propose a model for Csg in the following manner:
(14)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 (14) 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 optimum values of β1, β2, and β3, where the optimum values 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 cases A–E are also shown in Figures 6(a)–6(e), respectively, which demonstrate that the model constants β1, β2, and β3 remain greater than unity for all cases. This is found to be consistent with the realisability analysis by Hawkes and Cant [26]. Figures 6(a)–6(e) demonstrate that the optimum values of β1, β2, and β3 change with respect to Δ, which is also consistent with earlier findings [9]. Moreover, the optimum values of β1, β2, and β3 for a given value of Δ vary between cases considered here (see Figures 6(a)–6(e)). The optimum values of β1, β2, and β3 can also be parameterised in the same manner in which β5 is parameterised in (13i), (13ii), and (13iii). However, this is not presented here as the models given by (7)–(9) fail to capture the positive contribution of Csg for cases A–C. Moreover, the CSGCAND, CSGCANT, and CSGCHAR models do not capture the correct qualitative behaviour of Csg even when the optimum values of β1, β2, and β3 are used. The CSGCHAR model tends to overpredict the negative values of Csg towards the unburned gas side and this behaviour becomes more prominent with increasing filter size. Figure 7 shows 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 predict the correct magnitude of Csg for optimum values of β1 and β2, but they do not satisfactorily capture the qualitative behaviour of Csg and underpredict (overpredict) its magnitude towards the burned gas side (middle) of the flame brush. 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 for the major part of the flame brush for all cases, when optimum values of β4, β5, aΣ, and c* are used.
Variations of Csg conditionally averaged in bins of c~ across the flame brush for Δ=8Δm≈0.8δth (top row) and Δ=24Δm≈2.4δth (bottom row) for cases A (1st column), B (2nd column), C (3rd column), D (4th column), and E (5th column) along with the predictions of CSGCAND, CSGCANT, CSGCHAR, and CSGNEW.
4. Conclusions
The LES modelling of the curvature term (Sd∇·N→)¯sΣgen of the Σgen transport equation has been analysed using a simplified chemistry based DNS database of freely propagating statistically planar turbulent premixed flames with a range of different turbulent Reynolds numbers Ret. The variation of Ret is brought about by modifying Da and Ka independently from each other. 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, which arise from (Sr+Sn) and St=-2Dκm, respectively. Detailed physical explanations have been provided for the observed filter size 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. The performance of the new model for Csg has been found to be either better than or comparable to the performances of the existing models. It is worth noting that the present analysis has been carried out using a DNS database with moderate values of Ret in the absence of the effects of detailed chemistry and differential diffusion. 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.
NomenclatureArabicaΣ:
Model parameter for the newly proposed Csg1 model
CH:
Model parameter for Cant et al. [1] Model
Cmean:
Mean curvature term of flame surface density transport equation
Csg:
Subgrid curvature term of flame surface density transport equation
Csg1:
Subgrid curvature term component due to the combined reaction and normal diffusion components of displacement speed
Csg2:
Subgrid curvature term due to the tangential diffusion component of displacement speed
c:
Reaction progress variable
c*:
Model parameter for the newly proposed Csg1 model
D:
Progress variable diffusivity
Da:
Damköhler number
G(r→):
Gaussian Kernel used for filtering DNS data
Ka:
Karlovitz number
k~:
Subgrid turbulent kinetic energy
k1,k2,k3, and k4:
Model parameter for the newly proposed Csg1 model
Le:
Lewis number
M→,Mi:
Resolved flame normal vector and its ith component
m:
Model parameter for the newly proposed Csg1 model
N→,Ni:
Local flame normal vector and its ith component
n:
Model parameter for the newly proposed Csg2 model
Pr:
Prandtl number
Ret:
Turbulent Reynolds number
ReΔ:
Subgrid turbulent Reynolds number
r1,r2,r3, and r4:
Model parameter for the newly proposed Csg2 model
Sd:
Displacement speed
SL:
Laminar flame speed
Sn:
Normal diffusion component of displacement speed
Sr:
Reaction diffusion component of displacement speed
St:
Tangential diffusion component of displacement speed
T:
Nondimensional temperature
T^:
Instantaneous gas temperature (dimensional)
Tac:
Activation temperature
Tad:
Adiabatic flame temperature
T0:
Reactant temperature
tc:
Chemical time scale
tf:
Initial eddy turnover time
tsim:
Simulation time
ui:
ith component of nondimensional fluid velocity
u′:
Initial root-mean-square velocity fluctuation
uΔ′:
Subgrid scale turbulent velocity fluctuation
xi:
ith Cartesian coordinate
w˙:
Reaction rate of reaction progress variable.
GreekαN:
Resolution parameter of Cant et al. [1] and Candel et al. [2] models
αΣ:
Model parameter for the newly proposed Csg1 model
β:
Zel’dovich number
β1:
Model parameter for Candel et al. [2] model
β2:
Model parameter for Cant et al. [1] model
β3:
Model parameter for Charlette et al. [4] model
β4:
Model parameter for the newly proposed Csg1 model
β5:
Model parameter for the newly proposed Csg2 model
γ:
Ratio of specific heats
Δ:
Large eddy simulation filter width
Δm:
DNS mesh size
δth:
Thermal flame thickness
η:
Kolmogorov length scale
κm:
Flame curvature
μ:
Dynamic viscosity
μ0:
Dynamic viscosity of unburned gas
ΞΔ:
Wrinkling factor
ρ:
Density
ρ0:
Unburned gas density
Σgen:
Generalised flame surface density
τ:
Heat release parameter
τη:
Kolmogorov eddy turn-over time.
Symbols(⋯)¯:
LES filtering operation
(⋯)~:
LES Favre filtering operation
(⋯)¯s:
LES surface averaging operation.
AcronymsCSGCFM:
Subgrid curvature model proposed by Candel et al. [2]
CSGCPB:
Subgrid curvature model proposed by Cant et al. [1]
CSGCHAR:
Subgrid curvature model proposed by Charlette et al. [4]
CSGNEW:
Newly proposed subgrid curvature model
DNS:
Direct Numerical Simulation
FSD:
Flame Surface Density
LES:
Large Eddy Simulation
NSCBCs:
Navier Stokes Characteristic Boundary Conditions
TRZ:
Thin Reaction Zones.
Acknowledgment
The financial support by EPSRC, UK is gratefully acknowledged.
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