The performance of algebraic flame surface density (FSD) models has been assessed for flames with nonunity Lewis number (Le) in the thin reaction zones regime, using a direct numerical simulation (DNS) database of freely propagating turbulent premixed flames with Le ranging from 0.34 to 1.2. The focus is on algebraic FSD models based on a power-law approach, and the effects of Lewis number on the fractal dimension D and inner cut-off scale ηi have been studied in detail. It has been found that D is strongly affected by Lewis number and increases significantly with decreasing Le. By contrast, ηi remains close to the laminar flame thermal thickness for all values of Le considered here. A parameterisation of D is proposed such that the effects of Lewis number are explicitly accounted for. The new parameterisation is used to propose a new algebraic model for FSD. The performance of the new model is assessed with respect to results for the generalised FSD obtained from explicitly LES-filtered DNS data. It has been found that the performance of the most existing models deteriorates with decreasing Lewis number, while the newly proposed model is found to perform as well or better than the most existing algebraic models for FSD.
1. Introduction
Reaction rate closure based on flame surface density (FSD) is one of the most popular approaches to combustion modelling in turbulent premixed flames [1–11]. In the context of LES the generalised FSD (Σgen) is defined as follows [3–11]:
(1)Σgen=|∇c|¯.
where the overbar denotes the LES filtering operation. The reaction progress variable c may be defined in terms of a reactant mass fraction YR, for example, c=(YR0-YR)/(YR0-YR∝) such that c rises monotonically from zero in fresh reactants (subscript 0) to unity in fully burned products (subscript ∝).
In the context of LES, several models have been proposed for the wrinkling factor Ξ [12–16], which is often used in the context of thickened flame modelling [13, 14]. The wrinkling factor Ξ is closely related to Σgen according to [12–16]:(2a)Ξ=Σgen|∇c-|.
Often, Ξ is expressed in terms of a power-law expression [7, 9, 13, 14] Ξ=(η0/ηi)D-2 in which η0 and ηi are the outer and inner cut-off scales and D is the fractal dimension. This leads to a power-law expression for Σgen as:
(2b)Σgen=Ξ|∇c-|=(Δηi)D-2|∇c-|,where, for LES, the outer cut-off scale ηo is taken to be equal to the filter width Δ. According to Peters [17], ηi scales with the Gibson length scale LG=SL3/ε in the corrugated flamelets (CF) regime, and with the Kolmogorov length scale η=(ν3/ε)1/4 in the thin reaction zones (TRZ) regime. Here, SL is the unstrained laminar burning velocity, v is the kinematic viscosity in the unburned gas, and ε is the dissipation rate of turbulent kinetic energy. Experimental analyses by Knikker et al. [7] and Roberts et al. [18] indicated that ηi scales with the Zel’dovich flame thickness δZ=αT0/SL, where αT0 is the thermal diffusivity in unburned gases. A recent a priori DNS analysis [9] demonstrated that ηi scales with LG and η for the CF and TRZ regimes, respectively, as suggested by Peters [17]. However, ηi is also found to scale with thermal flame thickness δth in both the CF and TRZ regimes [9]. North and Santavicca [19] parameterised D in terms of the root-mean-square (rms) turbulent velocity fluctuation u' as: D=2.05/(u'/SL+1)+2.35/(SL/u'+1), whereas Kerstein [20] suggested that D increases from 2 to 7/3 for increasing values of u'/SL, where D=7/3 is associated with the material surface.
Since combustion is set to remain a major practical means of energy conversion for the foreseeable future, it has become necessary to find novel ways to reduce carbon emissions from relatively conventional combustion systems. One such approach is the use of hydrogen-blended hydrocarbon fuels in IC engines, aeroengines, and furnaces. Increased abundance of fast diffusing species such as H and H_{2} leads to significant effects of differential diffusion of heat and mass in hydrogen-blended flames [21, 22], whereas these effects are relatively weaker in conventional hydrocarbon flames [22, 23]. The differential rates of thermal and mass diffusion in premixed flames are often characterised by the Lewis number Le which is defined as the ratio of the thermal diffusivity to mass diffusivity (i.e., Le=αT/Dc). Assigning a global characteristic value of Le is not straightforward since many species with different individual values of Le are involved in actual combustion. Often the Lewis number of the deficient reactant species is used as the characteristic Le [21, 24–28] and this approach has been adopted here. It is worth noting that, to date, most FSD-based modelling has been carried out for unity Lewis number flames (e.g., [1–11]) and the effects of differential diffusion of heat and mass on the statistical behaviour of FSD have rarely been addressed [28]. More specifically the effects of Le on D and ηi have not yet been analysed in detail, or in the context of power-law FSD reaction rate models. Moreover, most algebraic models for Σgen have been proposed for the CF regime where the effects of Le are not accounted for. Thus, it is important to assess the performance of existing models for combustion in the TRZ regime with nonunity Lewis number.
The present study aims to bridge this gap in the existing literature. In this respect the main objectives of the work are as the following.
To understand the effects of Lewis number on D and ηi in the context of LES modelling.
To assess the performance of existing wrinkling factor-based algebraic models of FSD in the context of LES for flames with nonunity global Lewis number based on a priori DNS analysis.
To identify or develop a power-law-based algebraic model for FSD in the context of LES which is capable of predicting the correct behaviour of FSD even for nonunity Lewis number flames.
The rest of the paper is organised as follows. An overview of the different algebraic FSD models considered here are presented in the next section. This will be followed by a brief discussion of the numerical implementation. Following this, results will be presented and subsequently discussed. Finally the main findings will be summarised and conclusions will be drawn.
2. Overview of Power-Law-Based FSD Models
A model for Ξ suggested by Angelberger et al. [4] (FSDA model) can be written in terms of Σgen as follows:(3a)Σgen=[1+aΓ(uΔ′SL)]|∇c-|,
where a=1.0 is a model parameter, uΔ′=2k~Δ/3 is the subgrid turbulent velocity fluctuation, k~Δ=(uiui~-u~iu~i)/2 is the subgrid turbulent kinetic energy and Q~=ρQ¯/ρ- denotes the Favre-filtered value of a general quantity Q. In (3a), Γ is an efficiency function which is given by:
(3b)Γ=0.75exp[-1.2(uΔ′SL)-0.3]·(Δδz)2/3Weller et al. [12] also presented an algebraic model for Ξ, which can be recast in the form (FSDW model):
(4)Σgen=[1+2c~(Θ-1)]|∇c-|,
where Θ=1+0.62uΔ′/SLReη and Reη=uΔ′·η/ν with η and ρ0 denoting the Kolmogorov length scale and unburned gas density respectively. Colin et al. [13] proposed an algebraic model for Ξ, which can be expressed in terms of FSD (FSDC model) as:
(5)Σgen=[1+αΓ(uΔ′SL)]|∇c-|,
where Γ is given by (3b), α=β×2ln(2)/[3cms(Ret1/2-1)] with Ret=ρ0u'l/μ0, where μ0 is the unburned gas viscosity and l is the integral length scale, β=1.0 and cms=0.28. The FSDC model requires three input parameters, namely uΔ′/SL, Δ/δz, and Ret. Charlette et al. [14] reduced the input parameters to only uΔ′/SL and Δ/δz by using (FSDCH model):
(6)Σgen=(1+min[Δδz,ΓΔ(uΔ′SL)])β1|∇c-|,
with the efficiency function(7a)ΓΔ=[((fu-a1+fΔ-a1)-1/a1)-b1+fRe-b1]-1/b1,
where ReΔ=uΔ′Δ/ν and with model constants b1=1.4, β1=0.5, Ck=1.5, and functions a1, fu, fΔ, and fRe are defined by:
(7b)a1=0.60+0.20exp[-0.1uΔ′SL]-0.20exp[-0.01Δδz],fu=4(27110Ck)1/2(1855Ck)(uΔ′SL)2,fΔ={(27110Ckπ4/3)[(Δδz)4/3-1]}1/2,fRe=[955exp(-1.5Ckπ4/3ReΔ-1)]1/2ReΔ1/2. Knikker et al. [7] proposed a model for Σgen (FSDK model) as:
(8)Σgen=(Δηi)βk|∇c-|,
where the inner cut-off scale ηi is taken to be ηi=3δz and βk is estimated based on a dynamic formulation as βk=[log〈|∇c-|〉-log〈|∇c-^|〉]/logγ, where c-^ denotes the reaction progress variable at the test filter level γΔ. Fureby [16] proposed a model for Ξ which can be written in terms of Σgen (FSDF model) as:
(9)Σgen=[Γ(uΔ′SL)]D-2·|∇c-|,
where Γ is given by (3b), and D is specified according to the parameterisation D=2.05/(uΔ′/SL+1)+2.35/(SL/uΔ′+1) [19].
In the present study, the performance of each algebraic model described above is assessed with respect to Σgen obtained from DNS. There are three requirements for each model. Firstly, the volume-averaged value of Σgen represents the total flame surface area, and therefore this quantity should not change with Δ. Secondly, the model should be able to capture the correct variation of the averaged value of Σgen conditional on c- across the flame brush. Thirdly, the correlation coefficient between the modelled and actual values of Σgen should be as close to unity as possible in order to capture the effects of local strain rate and curvature on Σgen.
3. Numerical Implementation
For the purposes of the analysis, a DNS database of three-dimensional turbulent premixed flames has been generated using the compressible DNS code SENGA [29]. Until recently most combustion DNS was carried out either in three dimensions with simplified chemistry or in two dimensions with detailed chemistry due to the limitations of available computational power. Although it is now possible to carry out three-dimensional DNS with detailed chemistry, such computations remain extremely expensive [30] and are not practical for a parametric study as in the present case. Thus three-dimensional DNS with single-step Arrhenius type chemistry has been used in the present study in which the effects of Lewis number are to be investigated in isolation.
For the present DNS database, the computational domain is considered to be a cube of size 24.1δth×24.1δth×24.1δth, which is discretised using a uniform grid of 230×230×230. The grid spacing is determined by the flame resolution, and in all cases, about 10 grid points are kept within δth=(Tad-T0)/max|∇T^|L, where Tad,T0 and T^ are the adiabatic flame, unburned reactant and instantaneous dimensional temperatures respectively, and the subscript L is used to refer to unstrained planar laminar flame quantities. The boundaries in the direction of mean flame propagation are taken to be partially nonreflecting and are specified using the Navier Stokes Characteristic Boundary Conditions formulation [31], while boundaries in the transverse direction were taken to be periodic. A 10th order central difference scheme was used for spatial discretisation for internal grid points and the order of differentiation gradually decreases to a one-sided second-order scheme at non-periodic boundaries [29]. A low storage 3rd-order Runge-Kutta scheme [32] is used for time advancement. The turbulent velocity field is initialised by using a standard pseudo-spectral method [33], and the flame is initialised using an unstrained planar steady laminar flame solution.
The initial values of u'/SL and l/δth for all the flames considered here are shown in Table 1 along with the values of heat release parameter τ=(Tad-T0)/T0, Damköhler number Da=lSL/u'δth, Karlovitz number Ka=(u'/SL)3/2(lSL/αT0)-1/2 and turbulent Reynolds number Ret=ρ0u'l/μ0. For all cases Ka remains greater than unity, which indicates that combustion is taking place in the TRZ regime [17]. Standard values are taken for Prandtl number (Pr=0.7), ratio of specific heats (γG=CP/CV=1.4), and the Zel’dovich number (βZ=Tac(Tad-T0)/Tad2=6.0), where Tac is the activation temperature.
Initial values of the simulation parameters and nondimensional numbers relevant to DNS database
Case
Le
u′/SL
l/δth
τ
Ret
Da
Ka
A
0.34
7.5
2.45
4.5
47.0
0.33
9.92
B
0.6
7.5
2.45
4.5
47.0
0.33
9.92
C
0.8
7.5
2.45
4.5
47.0
0.33
9.92
D
1.0
7.5
2.45
4.5
47.0
0.33
9.92
E
1.2
7.5
2.45
4.5
47.0
0.33
9.92
List of initial simulation parameters and nondimensional numbers for the DNS database based on which the Ret dependence of D is parameterised
Case
u′/SL
l/δth
τ
Ret
Da
Ka
A1
5.0
1.67
4.5
22
0.33
6.54
B1
6.25
1.44
4.5
23.5
0.23
9.84
C1
7.5
2.5
4.5
49.0
0.33
9.84
D1
9.0
4.31
4.5
100.0
0.48
9.84
E1
11.25
3.75
4.5
110
0.33
14.73
In all cases, statistics were collected after three eddy turn-over times (i.e., 3tf=3l/u'), which corresponds to one chemical time scale (i.e., tc=δth/SL). The turbulent kinetic energy and its dissipation rate in the unburned reactants ahead of the flame were slowly varying at tsim=3.0l/u' and the qualitative nature of the statistics was found to have remained unchanged since t=2.0l/u' for all cases. By the time the statistics were extracted, the value of u'/SL in the unburned reactants ahead of the flame had decayed by about 50%, while the value of l/δth had increased by about 1.7 times, relative to their initial values. Further details on the flame-turbulence interaction of this DNS database may be found in [27, 28]. The present simulation time is short, but remains comparable to several studies [3, 8–10, 14, 34–37] which have contributed significantly to the fundamental understanding and modelling of turbulent premixed combustion in the past. The DNS data was explicitly filtered according to the integral Q(x→)¯=∫Q(x→-r→)G(r→)dr→ using a Gaussian kernel given by the expression G(r→)=(6/πΔ2)3/2exp(-6r→·r→/Δ2). The results will be presented for Δ ranging from Δ=4Δm≈0.4δth to Δ=24Δm≈2.4δth, where Δm is the DNS grid spacing (Δm≈0.1δth). These filter sizes are comparable to the range of Δ used in a priori DNS analysis in several previous studies [3, 8–10, 14], and span a useful range of length scales from Δ comparable to 0.4δth≈0.8δz, where the flame is partially resolved, up to 2.4δth≈4.8δz, where the flame becomes fully unresolved and Δ is comparable to the integral length scale. For these filter widths, the underlying combustion process ranges from the “laminar flamelets-G DNS” [38] combustion regime (for Δ=0.4δth≈0.8δz) to well within the TRZ regime (for Δ≥0.5δth≈δz) on the regime diagrams by Pitsch and Duchamp de Lageneste [38] and Düsing et al. [39]. However, these regime diagrams have been proposed based on scaling arguments for unity Lewis number flames and the likely effects of nonunity Lewis number on these regime diagrams have yet to be ascertained. This topic is the subject of a separate investigation and will not be taken up in this paper.
4. Results and Discussion4.1. Effects of Le on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M160"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M161"><mml:mrow><mml:msub><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
The power law expression (2b) for Σgen may be rewritten as:
(10)log[〈Σgen〉〈|∇c-|〉]=(D-2)logΔ-(D-2)log(ηi),
where the angled brackets indicate a volume-averaging operation. The variation of 〈Σgen〉/〈|∇c-|〉 with the ratio (Δ/δz) is shown in Figure 1 on a log-log plot for all the different Lewis number cases. The quantity 〈Σgen〉 denotes the total flame surface area which remains independent of filter size Δ. By contrast, the quantity 〈|∇c-|〉 denotes the resolved portion of the flame wrinkling, which decreases with increasing Δ. As a result, log[〈Σgen〉/〈|∇c-|〉] increases with increasing Δ. The variation of log[〈Σgen〉/〈|∇c-|〉] with log(Δ/δz) is linear when Δ≫δz but becomes nonlinear for Δ≪δz. The best-fit straight line representing the greatest slope of the linear variation has been used to obtain values of D and ηi. It has been found that ηi/δz remains independent of Le, and for all cases ηi remains on the order of thermal flame thickness δth (i.e., ηi/δth≈1.0), which is about twice the Zel’dovich flame thickness δz for the present thermochemistry (i.e., ηi=1.79δz≈δth). The scaling of the inner cut-off scale ηi with δz is consistent with previous DNS [9] and experimental [7, 18] findings. Figure 1 shows that the slope of the linear region decreases with increasing Lewis number (i.e., in moving from case a to case e), which suggests that the fractal dimension D decreases with increasing Le.
Variation of 〈Σgen〉/〈∇c-〉 with Δ/δz on a log-log plot for (a–e) cases A–E. Prediction of 〈Σgen〉/〈|∇c-|〉=(Δ/ηi)D-2 with ηi obtained from DNS and (D-2) according to (11) is also shown.
Le = 0.34
Le = 0.6
Le = 0.8
Le = 1.0
Le = 1.2
Contours of reaction progress variable c in the x1-x2 midplane are shown in Figure 2 for all cases and show that the extent of flame wrinkling is significantly greater at lower Lewis number. The rate of flame area generation increases with decreasing Le, and this behaviour is particularly noticeable for the cases with Le = 0.34 and Le = 0.6 because of the occurrence of thermo-diffusive instabilities [21, 24–28]. This can be substantiated from values of the ratio of turbulent to laminar flame surface area AT/AL obtained by volume integration of |∇c| (i.e., A=∫ϑ|∇c|dϑ). This produces the values AT/AL=3.93, 2.66, 2.11, 1.84, and 1.76 for the cases with Le = 0.34, 0.6, 0.8, 1.0, and 1.2, respectively, at the time when statistics were extracted. The experimental findings of North and Santavicca [19] suggested that D increases with increasing u'/SL~Ret1/4Ka1/2, which indicates that D is expected to have a dependence on both Ret and Ka. Moreover, the analysis of Kerstein [20] suggested that D is expected to assume an asymptotic value of 7/3 for large values of Ret and Ka. The present findings indicate that Le also has an influence on D in addition to Ret and Ka, and that D can assume values greater than 7/3 for flames with Le≪1.0 (see Figure 1). The Karlovitz number Ka dependence of D for unity Lewis number flames has been analysed in detail by Chakraborty and Klein [9] and they parameterised D as: D=2+(1/3)erf(2Ka), which does not account for the effects of Ret and Le. The parameterisation proposed by Chakraborty and Klein [9] has been extended here by accounting for the effects of Karlovitz number, turbulent Reynolds number, and global Lewis number (i.e., Ka, Ret, and Le) according to the following:
(11)D=2+13erf(3.0Ka)[1-exp(-0.1(RetAm)1.6)]Le-0.45,
where Am≈7.5 is a model parameter. Further details on the basis of this parameterisation are given in Appendix A.
Contours of c in the x1−x2 midplane at time t=δth/SL for (a–e) cases A–E.
The prediction of 〈Σgen〉/〈|∇c-|〉=(Δ/ηi)D-2 with ηi obtained from DNS and D obtained from (11) is also shown in Figure 1, which indicates that (11) satisfactorily captures the best-fit straight line corresponding to the power law. It is worth noting that Ret and Ka in (11) were evaluated for this purpose based on u'/SL and l/δth in the unburned reactants. However, in actual LES simulations, D needs to be evaluated based on local velocity and length scale ratios (i.e., uΔ′/SL and Δ/δz). Here uΔ′ is estimated from the subgrid turbulent kinetic energy as uΔ′=2k~Δ/3 following previous studies [12, 15, 16]. The local Karlovitz number KaΔ can be evaluated as KaΔ=CKa(kΔ/SL)3/2(δz/Δ)1/2, where CKa is a model parameter. Similarly, the local turbulent Reynolds number RetΔ can be evaluated using RetΔ=CRe(ρ0uΔ′Δ/μ0). The choice of model constants CKa=6.6 and CRe=4.0 ensures an accurate prediction of D for Δ≥ηi and yields the value of D obtained based on the global quantities according to (11).
Based on the observed behaviour of D and ηi, a power-law expression for Σgen is proposed here (model FSDNEW):
(12)Σgen=|∇c-|[(1-f)+f(Δηi)D-2],
where f is a bridging function which increases monotonically from zero for small Δ (i.e., Δ/δth→0 or Δ≪δth) to unity for large Δ (i.e., Δ≫ηi or Δ≫δth). Equation (12) ensures that Σgen approaches |∇c-|(Δ/ηi)D-2 for large Δ and at the same time Σgen approaches |∇c-| (i.e., limΔ→0Σgen=limΔ→0|∇c-|=|∇c|) for small Δ. It has been found that Σgen≈|∇c-| provides better agreement with Σgen obtained from DNS data for Δ≤0.8ηi, whereas the power-law Σgen=|∇c-|(Δ/ηi)D-2 starts to predict Σgen more accurately for Δ≥1.2ηi (see Figure 1). Based on this observation, the bridging function f is taken to be f=1/[1+exp{-60(Δ/ηi-1.0)}], which ensures a smooth transition between 0.8ηi<Δ<1.2ηi. As ηi is found to scale with δz (i.e., ηi≈1.79δz≈δth according to the present thermochemistry), ηi in (12) is taken to be the thermal flame thickness δth.
The performance of the various algebraic models for Σgen will be assessed next, using the model requirements stated earlier.
4.2. Performance of Models for the Volume-Averaged FSD <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M281"><mml:mrow><mml:mo stretchy="false">〈</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mtext>gen</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">〉</mml:mo></mml:mrow></mml:math></inline-formula>
The inaccuracy in the model predictions of 〈Σgen〉 can be characterised using a percentage error (PE):
(13)PE=〈Σgen〉model-〈Σgen〉〈Σgen〉×100,
where 〈Σgen〉model is the volume-averaged value of the model prediction of 〈Σgen〉. Results for the PE for a range of filter size Δ are shown Figure 3. These demonstrate that the models denoted by FSDA (see (3a) and (3b)) and FSDC (see (5)) overpredict 〈Σgen〉 for all the Lewis number cases, and that the level of overprediction increases with increasing Δ. The FSDW model (see (4)) also overpredicts 〈Σgen〉, although the level of overprediction decreases for Δ≫δth, especially for cases with Le≥0.6 (i.e., cases B–E). The FSDC model has greater PE than both the FSDA and FSDW models for all Δ in the same cases. However, the FSDW model has the highest PE relative to both the FSDA and FSDC models for all Δ in the Le=0.34 case.
Percentage error (13) of the model prediction from 〈Σgen〉 obtained from DNS for LES filter widths Δ=4Δm=0.4δth; Δ=8Δm=0.8δth; Δ=12Δm=1.2δth; Δ=16Δm=1.6δth; Δ=20Δm=2.0δth; Δ=24Δm=2.4δth for (a–e) cases A–E.
The FSDCH (6), FSDA, and FSDC models provide accurate predictions of 〈Σgen〉 at small values of Δ (i.e., Δ≪δz) but they overpredict 〈Σgen〉 for large values of Δ (i.e., Δ≫δz). The FSDF model (9) predicts accurately for small Δ, and marginally underpredicts for larger Δ, for cases with Le≥0.6. However, the FSDF model remains better than the FSDA, FSDC, FSDCH, and FSDW models. The FSDNEW model (12) provides an accurate prediction of 〈Σgen〉 for all filter sizes because this model is designed to do so for all values of Le. The PE for the FSDCH model remains small for cases with Le≈1.0 (i.e., cases C–E), although the FSDCH model overpredicts 〈Σgen〉 for Δ≫δth for cases with Le≪1 (i.e., cases A and B). The FSDK model (see (8)) underpredicts the value of 〈Σgen〉 for all Δ for all cases. However, the level of underprediction of the FSDK model decreases for larger Δ.
The PEs for the FSDF and FSDNEW models remain negligible in comparison to the PEs for all the other models. Note that Σgen should approach |∇c| (i.e., limuΔ′→0Σgen=limuΔ′→0|∇c|¯=|∇c|) when uΔ′ vanishes because the flow tends to be fully resolved (i.e., limΔ→0uΔ′=0 and limΔ→0Σgen=|∇c|). Although the FSDF model performs well for all Δ for all the cases considered here, Σgen does not tend to |∇c| as uΔ′ approaches zero, but instead predicts a finite value close to zero. This limitation of the FSDF model can be avoided using a modified form of (8) (MSFDF model):
(14)Σgen=|∇c-|[(1-f)+f(ΓuΔ′SL)D-2],
where f=1/[1+exp{-60(Δ/δth-1.0)}] is a bridging function as before, the efficiency function Γ is given by (3b) and D=2.05/(uΔ′/SL+1)+2.35/(SL/uΔ′+1) [19]. Equation (14) ensures that Σgen becomes exactly equal to |∇c| when the flow is fully resolved (i.e., Δ≪ηi or Δ→0), where uΔ′ also vanishes (i.e., limΔ→0uΔ′=0). Figure 3 shows that the modification given by (14) does not appreciably alter the performance of (8) while ensuring the correct asymptotic behaviour. Note that the parameterisation of D and Γ according to [19] and (3b), respectively, is essential for the satisfactory performance of the FSDF model. Using (13), for D in the FSDF model is found to lead to a deterioration in its performance. Similarly, using D as given by [19] in (12) worsens the performance of the FSDNEW model.
The FSDK model is based on the power-law Ξ=(η0/ηi)D-2 which is strictly valid only for filter sizes Δ which are sufficiently greater than ηi (i.e., Δ≫ηi), as can be seen from Figure 1. Hence, the predictive capability of the FSDK model improves when Δ>ηi (see Figure 3). However, the FSDK model underpredicts 〈Σgen〉 because the inner cut-off scale is taken to be 3δz in this model whereas ηi≈1.79δz for all the cases considered here. An accurate estimation of ηi in the framework of the FSDK model results in comparable performance to the FSDNEW model for large Δ (i.e., Δ≫ηi). Moreover, Σgen vanishes when Δ→0 according to the FSDK model, whereas Σgen should approach |∇c| when Δ→0 (i.e. limuΔ′→0Σgen=limuΔ′→0|∇c|¯=|∇c|). This limitation can be avoided by modifying the FSDK model in the same manner as shown in (14) for the FSDF model (not shown here for conciseness).
The stretch-rate K=(1/δA)d(δA)/dt=aT+Sd∇·N→ represents the fractional rate of change of flame surface area A [1], where Sd=Dc/Dt/|∇c| is the displacement speed, N→=-∇c/|∇c| is the local flame normal vector and aT=(δij-NiNj)∂ui/∂xj is the tangential strain rate. It is possible to decompose Sd into the reaction, normal diffusion and tangential diffusion components (i.e., Sr,Sn, and St) [8–10, 40, 41]:
(15)Sr=w˙ρ|∇c|,Sn=N→·∇(ρDcN→∇c)ρ|∇c|,St=-Dc∇·N→.
It has been shown in several previous studies [5, 6, 8, 10, 25] that (aT)¯s remains positive throughout the flame brush and thus acts to generate flame surface area, whereas the contribution of curvature to stretch (Sd∇·N→)¯s=[(Sr+Sn)∇·N→]¯s-[Dc(∇·N)2]¯s is primarily responsible for flame surface area destruction. The equilibrium of flame surface area generation and destruction yields (K)¯s=0, which gives rise to [9]:
(16)(aT)¯s=-[(Sr+Sn)∇·N→]¯s+[Dc(∇·N→)2]¯s.
The stretch rate induced by -[Dc(∇·N→)2]¯s becomes the leading order sink term in the thin reaction zones regime [8–10, 42]. However, most algebraic models (e.g., FSDA, FSDC, FSDCH, and FSDW) were proposed in the CF regime based on the equilibrium of the stretch rates induced by [(Sr+Sn)∇·N→]¯s and (aT)¯s, and the flame surface area destruction due to -[Dc(∇·N→)2]¯s was ignored [4, 12–14]. As a result, these models underestimate the flame surface area destruction rate in the thin reaction zones regime, which leads to overprediction of 〈Σgen〉 for the FSDA, FSDC, FSDCH, and FSDW models.
The disagreement between the FSDF model prediction and DNS data originates principally due to the inaccuracy in estimating Γ and D, while the difference between the FSDK prediction and DNS data arises from inaccurate estimation of ηi. Hence a more accurate estimation of Γ, D, and ηi will result in better performance of both the FSDF and FSDK models.
4.3. Performance of Models for the Variation of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M383"><mml:mrow><mml:msub><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mtext>gen</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
It is important to assess the models based on their ability to capture the correct variation of Σgen with c- across the flame brush. The variation of mean Σgen conditionally averaged on c- is shown in Figure 4 for Δ=8Δm=0.8δth and Figure 5 for Δ=24Δm=2.4δth, respectively. These filter widths have been chosen since they correspond to Δ<ηi and Δ>ηi respectively. The following observations can be made from Figure 4 about the model predictions at Δ=8Δm=0.8δth.
The models FSDA, FSDC, FSDCH, FSDF, and FSDNEW tend to capture the variation of the conditional mean value of Σgen with c- obtained from DNS data. The prediction of the MFSDF model remains comparable to that of the FSDF model for Δ=8Δm=0.8δth.
The FSDW model consistently overpredicts the conditional mean value of Σgen for all cases. The FSDW model also predicts a skewed shape, which fails to capture the trend predicted by DNS.
The model FSDK underpredicts the conditional mean value of Σgen in all cases. The physical explanations provided earlier for the underprediction of 〈Σgen〉 by the FSDK model is also responsible for the underprediction seen here.
Variation of mean values of Σgen×δz conditional on c- across the flame brush for Δ=8Δm=0.8δth according to DNS, FSDA, FSDC, FSDW, FSDCH, FSDK, FSDF, FSDNEW, and MFSDF predictions for (a–e) cases A–E.
Variation of mean values of Σgen×δz conditional on c- across the flame brush for Δ=24Δm=2.4δth according to DNS, FSDA, FSDC, FSDW, FSDCH, FSDK, FSDF, FSDNEW, and MFSDF predictions for (a–e) cases A–E.
A comparison between Figures 4 and 5 reveals that the predictions of the various algebraic FSD models exhibit greater spread for Δ=24Δm=2.4δth than in the case of Δ=8Δm=0.8δth. The following observations can be made from Figure 5 about the model predictions at Δ=24Δm=2.4δth.
Similar to Δ=8Δm, the FSDW model predicts a peak at c-> 0.6, whereas the peak value of conditionally averaged Σgen from DNS occurs at c-≈ 0.5 for all the cases.
The models FSDW, FSDA, FSDC, and FSDCH tend to overpredict the conditionally averaged value of Σgen and the level of the overprediction increases with decreasing Lewis number.
The models FSDF, FSDK, FSDNEW, and MFSDF tend to predict the conditionally averaged value of Σgen satisfactorily throughout the flame brush.
The difference in the predictions of the models MFSDF, and FSDF seem to be very small for all the flames considered here.
The inaccuracy in the predictions of the mean value of Σgen conditional on c- can be characterised once again using a percentage error (PE_{2}):
(17)PE2=ΣcondMODEL-ΣcondDNSΣcondmax×100,
where ΣcondMODEL and ΣcondDNS are the mean values of Σgen conditional on c- as obtained from model prediction and DNS respectively, and Σcondmax is the maximum value of conditionally averaged Σgen obtained from DNS. The error in the model prediction according to (16) is shown in Figure 6 for filter size Δ=8Δm=0.8δth and in Figure 7 for filter size Δ=24Δm=2.4δth. Note that the models predicting PE2 outside a margin of ±15% have been discarded. In the case of Le = 0.34 (case A) the models FSDNEW, FSDF, MFSDF, and FSDC stay within the ±15% error limit for Δ=8Δm whereas only the models FSDF, MFSDF, FSDK and FSDNEW remain within the ±15% error limit for Δ=24Δm. As Le increases to 0.6 (case B), the models FSDNEW, FSDCH, FSDF, MFSDF, FSDC, FSDA, and FSDK predict within the ±15% error margin and have been listed in terms of decreasing accuracy for Δ=8Δm. For case B only the predictions of FSDNEW, FSDF, MFSDF and FSDK remain within the ±15% error margin for Δ=24Δm. In the Le = 0.8 case (case C), the models FSDF, FSDNEW, MFSDF, FSDCH, FSDA, FSDC, FSDK and FSDW all provide predictions within ±15% for Δ=8Δm, whereas the predictions of FSDNEW, FSDF, MFSDF, FSDCH, FSDK and FSDW remain within ±15% for Δ=24Δm. For Le = 1.0 and 1.2 (cases D and E) the models FSDF, MFSDF, FSDNEW, FSDCH, FSDA, FSDC, FSDK and FSDW all predict within the ±15% error margin for Δ=8Δm, while the models FSDF, MFSDF, FSDNEW, FSDK and FSDCH predict within ±15% for Δ=24Δm. The model FSDW was found to predict within the ±15% error margin for Δ=24Δm in the Le = 1.0 flame but its prediction remains marginally beyond the ±15% error margin for Δ=24Δm for the Le = 1.2 flame considered here (The maximum magnitude of PE_{2} for the FSDW model in the Le = 1.2 case is 15.2%, and the variation of PE_{2} with c- in this case is qualitatively similar to the Le = 1.0 case considered here).
Variation of percentage error (17) on c- across the flame brush for Δ=8Δm=0.8δth according to FSDA, FSDC, FSDW, FSDCH, FSDK, FSDF, FSDNEW, and MFSDF predictions for (a–e) cases A–E.
Variation of percentage error (17) on c- across the flame brush for Δ=24Δm=2.4δth according to FSDA, FSDC, FSDW, FSDCH, FSDK, FSDF, FSDNEW, and MFSDF predictions for (a–e) cases A–E.
Comparing the performance of the models at Δ=8Δm and Δ=24Δm, it can be seen that FSDA, FSDCH and FSDC predict Σgen satisfactorily at Δ=8Δm but the agreement with DNS deteriorates at Δ=24Δm. By contrast, the FSDK prediction is closer to DNS data at Δ=24Δm than at Δ=8Δm. The models FSDF, MFSDF, and FSDNEW fare well at both Δ=8Δm and Δ=24Δm for all the Lewis number values considered here. It is worth noting that the FSDNEW model was designed to predict the volume-averaged value of generalised FSD 〈Σgen〉, but judging from Figures 4–7, this model also performs satisfactorily with respect to predicting the correct variation of Σgen across the flame brush.
The prediction of the model FSDK improves with increasing filter width Δ, unlike the other models, which is consistent with observations made in the context of Figure 3. The prediction of the FSDW model remains skewed towards the product side of the flame brush due to the c~ dependence of Ξ (i.e., Ξ=1+1.24c~uΔ′/SLReη) proposed in [12]. The FSDW, FSDA, FSDC, and FSDCH models underestimate the destruction rate of flame surface area in the thin reaction zones regime due to the underestimation of FSD destruction arising due to the curvature stretch contribution -[Dc(∇·N→)2]¯s, which eventually leads to the overprediction of conditionally averaged value of Σgen.
4.4. Performance of Models for the Local <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M458"><mml:mrow><mml:msub><mml:mrow><mml:mi>Σ</mml:mi></mml:mrow><mml:mrow><mml:mtext>gen</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> Behaviour
The FSD predicted by the models should have the correct resolved strain rate and curvature dependence in the context of LES and thus the correlation coefficient between the FSD obtained from DNS and from the model prediction should remain as close to unity as possible. The variation of the correlation coefficients between the model prediction and generalised FSD Σgen obtained from DNS in the range of filtered reaction progress variable 0.1≤c-≤0.9 are shown in Figure 8 for different filter widths. The regions corresponding to 0.1<c- and c->0.9 have been ignored since the correlation coefficients have little physical significance in these regions due to the small values of Σgen obtained from both DNS and model predictions. Figure 8 indicates that the correlation coefficients decrease with increasing Δ due to increased unresolved subgrid wrinkling, which makes the local variation of Σgen different from |∇c-|. The extent of the deviation of the correlation coefficients from unity increases with decreasing Le for a given value of Δ. Figure 8 indicates that the models FSDA, FSDC, FSDCH, FSDF, MFSDF, FSDK, FSDNEW, and FSDW have comparable correlation coefficients, which deviate considerably from unity for large values of Δ. This indicates that algebraic models may not be able to predict FSD such that its local strain rate and curvature dependencies can be appropriately captured, especially in the TRZ regime. Hence a transport equation for FSD might need to be solved to account for the local strain rate and curvature effects on Σgen [5, 6, 8, 10, 11].
Correlation coefficients between the modelled and the actual values of Σgen in the c- range 0.1≤c-≤0.9 for filter widths Δ=4Δm=0.4δth; Δ=8Δm=0.8δth; Δ=12Δm=1.2δth; Δ=16Δm=1.6δth; Δ=20Δm=2.0δth; Δ=24Δm=2.4δth for (a–e) cases A–E.
5. Conclusions
The performance of several wrinkling factor based LES algebraic models for Σgen has been assessed for nonunity Lewis number flames in the TRZ regime based on a DNS database of freely propagating statistically planar turbulent premixed flames with Le ranging from 0.34 to 1.2. It has been found that the fractal dimension D increases with decreasing Le, whereas Le does not have any significant influence on the value of the normalised inner cut-off scale ηi/δz. For all Lewis number cases the inner cut-off scale is found to be equal to the thermal flame thickness (i.e., ηi≈δth). Based on the analysis of DNS data, a new parameterisation of D is proposed, where the effects of Le are explicitly accounted for. This new parameterisation of D has been used to propose a power-law based model for Σgen to account for nonunity Lewis number effects. The performance of this new model has been assessed with respect to Σgen obtained from DNS data alongside other existing models. The new model was found to be capable of predicting the behaviour of Σgen in the TRZ regime with greater or comparable accuracy in comparison to the existing models for all values of Le considered here. However, the present study has been carried out for moderate values of turbulent Reynolds number Ret and the effects of detailed chemistry and transport are not accounted for. Thus, three-dimensional DNS with detailed chemistry will be necessary, together with experimental data, for a more comprehensive assessment of LES algebraic models for Σgen.
AppendixA. Effects of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M490"><mml:mrow><mml:msub><mml:mrow><mml:mi>Re</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> on Fractal Dimension <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M491"><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>
The effects of Ret on D have been analysed based on a simplified chemistry based DNS database [43, 44], in which the variation of Ret~Da2Ka2 is brought about by modifying Da and Ka independently of each other. The initial values of u'/SL and l/δth for all the flames in this DNS database are shown in Table 1(a) along with the values of heat release parameter τ=(Tad-T0)/T0, Damköhler number Da=lSL/u'δth, Karlovitz number Ka=(u'/SL)3/2(lSL/αT0)-1/2, and turbulent Reynolds number Ret=ρ0u'l/μ0.
The variations of log(〈Σgen〉/〈|∇c-|〉) with log(Δ/δz) for cases A1–E1 are shown in Figure 9, which demonstrate that D is greater for flames with higher Ret, and that D attains an asymptotic value of 7/3 for unity Lewis number flames with high values of Ret (e.g., cases D1 and E1). The prediction of 〈Σgen〉/〈|∇c-|〉=(Δ/ηi)D-2 with ηi obtained from DNS and D obtained from (11) is also shown in Figure 9, which indicates that (11) satisfactorily captures the slope of the best-fit straight line.
Variation of 〈Σgen〉/〈|∇c-|〉 with Δ/δz on a log-log plot for (a–e) cases A1–E1. The prediction of 〈Σgen〉/〈|∇c-|〉=(Δ/ηi)D-2 with ηi obtained from DNS and (D-2) according to (11) is also shown.
Acknowledgment
The authors are grateful to EPSRC, UK, for financial assistance.
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