A direct numerical simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different turbulent Reynolds numbers has been used to assess the performance of algebraic flame surface density (FSD) models based on a fractal representation of the flame wrinkling factor. The turbulent Reynolds number Re_{t} has been varied by modifying the Karlovitz number Ka and the Damköhler number Da independently of each other in such a way that the flames remain within the thin reaction zones regime. It has been found that the turbulent Reynolds number and the Karlovitz number both have a significant influence on the fractal dimension, which is found to increase with increasing Re_{t} and Ka before reaching an asymptotic value for large values of Re_{t} and Ka. A parameterisation of the fractal dimension is presented in which the effects of the Reynolds and the Karlovitz numbers are explicitly taken into account. By contrast, the inner cut-off scale normalised by the Zel’dovich flame thickness ηi/δz does not exhibit any significant dependence on Re_{t} for the cases considered here. The performance of several algebraic FSD models has been assessed based on various criteria. Most of the algebraic models show a deterioration in performance with increasing the LES filter width.
1. Introduction
Large eddy simulation (LES) is becoming increasingly popular for computational fluid dynamics (CFD) analysis of turbulent reacting flows due to the advancement and increased affordability of high-performance computing. The exponential temperature dependence of the chemical reaction rate poses one of the major challenges in LES modelling of turbulent reacting flows [1, 2]. Reaction rate closure models based on the concept of flame surface density (FSD) are well established in the context of the Reynolds averaged Navier Stokes simulations [3, 4] of turbulent premixed flames. However, the application of FSD-based modelling in LES is relatively recent [5–10]. The generalised FSD (Σgen) is defined as [5]
(1)Σgen=|∇c|¯,
where c is the reaction progress variable. The overbar indicates the LES filtering operation in which the filtered value Q- of a general quantity is evaluated as Q-(x→)=∫Q(x→-r→)G(r→)dr→, where G(r→) is a suitable filter function [5]. The combined contribution of the filtered reaction and molecular diffusion rates w˙+∇·(ρDc∇c)¯ can be modelled using Σgen as w˙+∇·(ρDc∇c)¯=(ρSd)¯sΣgen, where ρ is the fluid density, Dc is the progress variable diffusivity, (ρSd)¯s is the density-weighted surface-filtered displacement speed Sd=(Dc/Dt)/|∇c|, and (Q)¯s=Q|∇c|¯/|∇c|¯ is the surface-filtered value of a general quantity Q. Often, Σgen is expressed in terms of the wrinkling factor Ξ, which is defined as
(2)Ξ=|∇c|¯|∇c-|=Σgen|∇c-|.
Hence the prediction of Σgen depends on the accuracy of the modelling of Ξ [8]. Several models for Ξ have been proposed in the context of LES [11–16], many of which make use of a power-law scaling involving a fractal representation of the flame surface.
To date, most of the algebraic models for FSD based on the wrinkling factor have been developed for the corrugated flamelet (CF) regime [17] in which the flame thickness remains smaller than the Kolmogorov length scale. However, there has been no detailed assessment of the performance of these models in the thin reaction zone (TRZ) regime [17], in which the Kolmogorov length scale remains smaller than the flame thickness. Moreover, the effects of turbulent Reynolds number Ret on these models also remain to be investigated in detail. In order to address these gaps in the existing literature, the performance of several algebraic FSD models has been assessed here for TRZ regime combustion based on a direct numerical simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with different values of Ret. The variation of Ret is brought about by varying the Damköhler number Da and the Karlovitz number Ka independently of each other, using the following relation [17]:
(3)Ret~Da2Ka2~(u′SL)2Da~(u′SL)4Ka-2.
Here, Ret=ρ0u′l/μ0 is the turbulent Reynolds number, Da=lSL/u′δth is the Damköhler number and Ka=(u′/SL)3/2(lSL/αT0)-1/2 is the Karlovitz number [17]. Subscript 0 denotes quantities evaluated in the unburned reactants, u′ is the turbulent velocity fluctuation magnitude, l is the turbulence integral length scale, μ is the dynamic viscosity, αT is the thermal diffusivity, SL is the laminar burning velocity, and δth is the thermal thickness of the laminar flame.
The main objectives of the present study are
to assess the performance of a range of wrinkling factor based algebraic Σgen models in the TRZ regime,
to identify the influence of Ret on the performance of these models in the TRZ regime.
A summary of the existing wrinkling factor models will be provided in the next section. This will be followed by a discussion on the numerical implementation. Following this, the results will be presented and discussed. Finally, the main findings will be summarised and conclusions will be drawn.
2. Mathematical Background
The wrinkling factor Ξ can be expressed in the form of a power law [11–16, 18] Ξ=(η0/ηi)D-2, where η0 and ηi are the outer and inner cut-off scales and D is the fractal dimension of the flame surface. For LES, ηo is taken to be equal to the filter width Δ. According to Peters [17], ηi scales with the Gibson scale LG=SL3/ε (the Obukhov-Corrsin scale ηOC=(Dc3/ε)1/4) in the CF (TRZ) regime, with ε being the dissipation rate of turbulent kinetic energy (TKE). Knikker et al. [16] found by experiment that ηi scales with the Zel’dovich flame thickness δz=αT0/SL. A recent DNS analysis [8] found that ηi does indeed scale with LG and ηOC for the CF and TRZ regimes, respectively, but also scales with δz for both regimes.
North and Santavicca [19] parameterised D as D=2.05/(u′/SL+1)+2.35/(SL/u′+1), which suggests that D increases with u′/SL. Kerstein [20] indicated that D increases from 2 to 7/3 for increasing values of u′/SL, where D=7/3 is associated with a nonpropagating material surface. The a priori DNS analysis by Chakraborty and Klein [8] demonstrated that D indeed increases from a value slightly greater than 2.0 in the CF regime to 7/3 for high Ka unity Lewis number flames in the TRZ regime.
Weller et al. [11] proposed a model for Ξ, (denoted here as FSDW), which can be recast in terms of Σgen as
(4)Σgen=[1+2c~(Θ-1)]|∇c-|,
where Θ=1+0.62uΔ′/SLReη and Reη=ρ0uΔ′η/μ0 with η being the Kolmogorov length scale. The subgrid scale velocity fluctuation magnitude is defined as uΔ′=(uiui~-u~iu~i)/3. A model for Ξ proposed by Angelberger et al. [12] (denoted as FSDA) can be written as (5i)Σgen=[1+aΓ(uΔ′SL)]|∇c-|,
where a is a model parameter of the order of unity and the efficiency function Γ is defined as
(5ii)Γ=0.75exp[-1.2(uΔ′SL)-0.3](Δδz)2/3.Colin et al. [13] proposed a slightly different model (denoted as FSDC):
(6)Σgen=[1+αΓ(uΔ′SL)]|∇c-|,
where Γ is given by (5ii) and α=2ln(2)/[3cms(Ret1/2-1)] with cms=0.28. Charlette et al. [14] reduced the input parameters to only uΔ′/SL and Δ/δc using the expression (denoted as FSDCH)(7i)Σgen=(1+min[Δδc,ΓΔ(uΔ′SL)])β1|∇c-|
with the efficiency function
(7ii)ΓΔ=[((fu-a1+fΔ-a1)-1/a1)-b1+fRe-b1]-1/b1,
where δc=4.0μ0/ρ0SL, ReΔ=4(uΔ′/SL)·(Δ/δc) and with model constants b1=1.4, β1=0.5, Ck=1.5 and functions a1, fu, fΔ, and fRe defined by
(7iii)a1=0.60+0.20exp[-0.1uΔ′SL]-0.20exp[-0.01Δδc],fu=4(27110Ck)1/2(1855Ck)(uΔ′SL)2,fΔ={(27110Ckπ4/3)[(Δδc)4/3-1]}1/2,fRe=[955exp(-1.5Ckπ4/3ReΔ-1)]1/2ReΔ1/2.Fureby [15] proposed a model (denoted as FSDF) which can be written as:
(8)Σgen=(ΓuΔ′SL)D-2|∇c-|,
where Γ is given by (5ii) and D=2.05/(uΔ′/SL+1)+2.35/(SL/uΔ′+1) [19]. Knikker et al. [16] proposed a model (denoted as FSDK) as
(9)Σgen=(Δηi)βk|∇c-|,
where ηi=3δz and βk is dynamically evaluated as βk=[log〈|∇c-|〉^-log〈|∇c-^|〉]/logγ where c-^ denotes the filtered c at the test filter level γΔ. The angled bracket 〈⋯〉 indicates a volume averaging operation, as often used in dynamic models [16, 21].
In Section 4, the performance of these models is assessed with respect to Σgen obtained from DNS based on the following criteria.
Criterion 1.
As FSD represents the flame surface area to volume ratio [3], the volume-averaged value of the generalised FSD over the DNS domain 〈Σgen〉 represents the total flame surface area within the domain and therefore should be independent of Δ. Thus the model predictions of 〈Σgen〉 should not change with Δ.
Criterion 2.
The models for Σgen should be able to capture the correct variation of the mean values of Σgen conditional on c- across the flame brush.
Criterion 3.
The correlation coefficient between the modelled and actual values of Σgen should be as close as possible to unity, in order to capture correctly the local strain rate and the curvature effects on Σgen in the context of LES.
3. Numerical Implementation
Three-dimensional DNS of freely propagating statistically planar turbulent premixed flames has been carried out using the DNS code SENGA [22]. The 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. The grid spacing was determined by the flame resolution, and in all cases, about 10 grid points are kept within δth. The boundaries in the direction of the mean flame propagation (i.e., x1-direction) were taken to be partially nonreflecting whereas the transverse directions were taken to be periodic. Higher order finite-difference and Runge-Kutta schemes were employed for spatial discretisation and time advancement, respectively. The scalar fields were initialised using a steady unstrained planar laminar premixed flame solution. For the present study, standard values were taken for the Prandtl number (Pr=0.7), the Zel’dovich number (βZ=Tac(Tad-T0)/Tad2=6.0), and the ratio of specific heats (γg=CP/CV=1.4), where Tac,CP, and CV are the activation temperature and the specific heats at constant pressure and constant volume, respectively. The initial values of u′/SL, l/δth, Da, Ka, and Ret are listed in Table 1. In cases A, C, and E (B, C, and D), the values of u′/SL and l/δth were chosen to vary Ret by changing Ka (Da) while keeping Da(Ka) constant (see (3)). The heat release parameter τ=(Tad-T0)/T0 and the Lewis number Le are taken to be equal to 4.5 and 1.0, respectively. Table 1 shows that Ka remains greater than unity for all cases indicating the TRZ regime combustion [17]. In all cases the flame-turbulence interaction takes place under decaying turbulence, and the simulation time corresponds to one chemical time scale tc=δth/SL. This time is equal to 2.0tf in case D, 3.0tf in cases A, C, and E, and 4.34tf for case B where tf=l/u′ is the initial eddy turn-over time. The present simulation time remains comparable with several previous DNS studies which focused on the modelling of Σgen [5, 7–9, 14, 21, 23]. By the time statistics were extracted, the global TKE and volume-averaged burning rate were no longer changing rapidly with time [24]. The global level of turbulent velocity fluctuation had decayed by 53%, 61%, 45%, 24%, and 34% in comparison to the initial values for cases A–E, respectively. By contrast, the integral length scale increased by factors of 1.5 to 2.25, ensuring that a sufficient number of turbulent eddies was retained in each direction. Values for u′/SL, l/δth, and δth/η at the time when statistics were extracted were presented in [24, Table 2] and are not repeated here (note that the Karlovitz number was defined in [24] in terms of the thermal flame thickness δth as Ka=(u'/SL)1.5(l/δth)-1/2, whereas Ka in this paper is defined in terms of the Zel’dovich flame thickness δz=αT0/SL as Ka=(u′/SL)1.5(lSL/αT0)-1/2).
List of initial simulation parameters and nondimensional numbers.
Case
u'/SL
l/δth
τ
Ret
Da
Ka
A
5.0
1.67
4.5
22
0.33
6.54
B
6.25
1.44
4.5
23.5
0.23
9.84
C
7.5
2.5
4.5
49.0
0.33
9.82
D
9.0
4.31
4.5
100.0
0.48
9.83
E
11.25
3.75
4.5
110
0.33
14.73
In the present analysis, the DNS data is explicitly filtered using G(r→)=(6/πΔ2)3/2exp(-6r→·r→/Δ2) [5] to obtain the relevant filtered quantities. The results will be presented 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 [5, 7, 9, 14, 18] 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). For this range of filter widths, the underlying combustion process varies from the “laminar flamelets-G DNS” [25] combustion regime (for Δ=0.4δth≈0.7δz) to well within the TRZ regime (for Δ≥0.5δth≈δz) on the regime diagrams by Pitsch and Duchamp de Lageneste [25] and Düsing et al. [26].
4. Results and Discussion4.1. Flame Turbulence Interaction
Contours of c in the x1-x2 mid-plane at time tsim=1.0δth/SL are shown in Figures 1(a)–1(e) for cases A–E, respectively. Figure 1 shows that the extent of flame wrinkling increases with increasing u′/SL~Ret1/2/Da1/2~Ret1/4Ka1/2 (see Table 1). Moreover, the contours of c representing the preheat zone (i.e., c<0.5, see colour scale) are much more distorted than those representing the reaction zone (i.e., 0.7<c<0.9). This tendency becomes more prevalent with the increase in the Karlovitz number Ka, since the scale separation between δth and η increases with increasing Ka, allowing more energetic eddies to enter into the preheat zone causing greater distortion of the flame.
Contours of c in the x1-x2 midplane at time t=δth/SL for: (a–e) cases A–E.
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The power-law Ξ=Σgen/|∇c-|=(η0/ηi)D-2 produces the expression
(10)log[〈Σgen〉〈|∇c-|〉]=(D-2)log(Δηi),
where the angled brackets indicate the volume-averaging operation. The quantity 〈|∇c-|〉 decreases with increasing Δ due to an increase in the proportion of flame wrinkling that occurs at the sub-grid scales with increasing Δ. By contrast, 〈Σgen〉 indicates the total flame surface area in the computational domain, thus remaining independent of Δ. As a result, log(〈Σgen〉/〈|∇c-|〉) increases with increasing Δ, which can be substantiated from Figures 2(a)–2(e). The variation of log(〈Σgen〉/〈|∇c-|〉) with log(Δ/δz) is linear when Δ≫δz but becomes nonlinear for Δ≪δz. The slope of the best-fit straight line representing the greatest slope of the variation of log(〈Σgen〉/〈|∇c-|〉) with log(Δ/δz) gives D while the intersection point of this straight line with the abscissa gives logηi. It has been found that ηi/δz remains about 2.0 for all cases (i.e., ηi/δz≈1.785) and that δth=1.785δz for the present thermochemistry. This is qualitatively consistent with previous experimental [16, 27] and computational [8] findings.
Variation of 〈Σgen〉/〈|∇c-|〉 () with Δ/δz on a log-log plot for (a–e) cases A–E. The prediction of 〈Σgen〉/〈|∇c-|〉=(Δ/ηi)D-2 () with ηi obtained from DNS and D according to (11) () is also shown.
Figures 2(a)–2(e) demonstrate that D is greater for flames with higher Ret and that D attains an asymptotic value of 7/3 for flames with Ret≥50 (e.g., cases C, D, and E). The extent of the flame wrinkling and the flame surface area generation increases with increasing u′/SL. This can be substantiated from the values of normalised flame surface area AT/AL obtained by volume integrating |∇c| (i.e., ∫V|∇c|dV, where dV is an infinitesimal volume element), which yields AT/AL= 1.15, 1.33, 1.87, 3.63, and 3.70 for cases A–E, respectively, at the time when statistics were extracted. This behaviour is consistent with Figure 1 which demonstrates that the wrinkling of c isosurfaces increases with increasing u′/SL~Ret1/4Ka1/2~Ret1/2/Da1/2. Figure 2 suggests that D is expected to increase with increasing Ret~Da2Ka2 before assuming an asymptotic value when either Da or Ka is held constant. This behaviour is also qualitatively similar to the trend predicted by the parameterisation of North and Santavicca [19]. The parameterisation by Chakraborty and Klein [8], that is, D=2+1/3erf (2Ka), predicts D=7/3 for all the cases considered here because this parameterisation accounts for the dependence of D on Ka only. Based on the above findings, D is parameterised here as
(11)D=2+13erf(3Ka)[1-exp(-0.1(RetAm)1.6)],
where Am≈7.5 is a model parameter. The prediction of 〈Σgen〉/〈|∇c-|〉=(Δ/ηi)D-2 with ηi obtained from DNS and D obtained from (11) is also shown in Figures 2(a)–2(e), which indicates that (11) satisfactorily captures the best-fit straight line corresponding to the power law. According to (11), D increases from a value close to 2 for small values of u′/SL (e.g., cases A and B) to an asymptotic value of 7/3 for large values of Ret and Ka (e.g., cases D and E), according to Kerstein [20].
It has been found in previous analyses [8, 19, 28] that D approaches 7/3 for unity Lewis number flames within the flamelet regime when Ret approaches a value of about 50 and the Karlovitz number remains greater than unity (i.e., Ka>1). The fractal dimension for the present case C is found to be D=2.31, and this small deviation from 7/3=2.33 is within the statistical noise and should not be over-interpreted. The fractal dimensions for cases D and E are equal to 7/3 which indicates that for high Karlovitz numbers (i.e., Ka>1) within the flamelet regime, D approaches an asymptotic value of 7/3 for Ret≥50 according to the present analysis.
Note that Ret and Ka in (11) were evaluated here 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 sub-grid TKE k~Δ=(uiui~-u~iu~i)/2 (i.e., uΔ′=2k~Δ/3) following previous studies [6, 8, 10–12, 15, 18]. 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.
A power-law-based expression for Σgen is proposed here based on the observed behaviours of D and ηi in Figure 2:
(12)Σgen=|∇c-|[(1-f)+f(Δηi)D-2],
where f is a bridging function which increases monotonically from 0 for small Δ (i.e., Δ/δth→0 or Δ≪δth) to 1 for large Δ (i.e., Δ≫ηi or Δ≫δth). The expression given by (12) will be denoted as FSDNEW. Equation (12) ensures that Σgen approaches |∇c-|(Δ/ηi)D-2 (i.e., Σgen=|∇c-|(Δ/ηi)D-2) for large Δ (i.e., Δ≫ηi or Δ≫δth) and at the same time Σgen approaches |∇c-| (i.e., limΔ→0Σgen=limΔ→0|∇c-|=|∇c|) for small Δ (i.e., Δ/δth→0). It has been found from Figure 2 that Σgen≈|∇c-| provides better agreement with Σgen obtained from the 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. 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 0.8ηi<Δ<1.2ηi. As ηi is found to scale with δz (i.e., ηi=δth=1.785δz according to the present thermochemistry), ηi in (12) is taken to be the thermal flame thickness δth. In the context of LES, D needs to be evaluated based on local quantities, which is achieved by replacing Ka and Ret in (11) with their local values KaΔ=CKa(k~Δ/SL)3/2(Δ/δz)-1/2 and RetΔ=CRe(ρ0uΔ′Δ/μ0), respectively. 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).
4.3. Assessment of Model PerformanceCriterion 1.
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 model prediction of 〈Σgen〉. Results for the PE for a range of Δ are shown in Figure 3, which demonstrate that the models denoted by FSDA (see (5i) and (5ii) and FSDC (see (6)) overpredict 〈Σgen〉 for all the cases and that the level of overprediction increases with increasing Δ. The FSDW model (see (4)) also overpredicts the value of 〈Σgen〉, although the level of overprediction for Δ≫δth is smaller for this model, especially for the cases with higher Ret (i.e., cases D and E). The FSDC model remains inferior to both the FSDA and FSDW models for all Δ in all cases. The PE for the FSDCH model (see (7i), (7ii), and (7iii) remains small for the small and moderate Ret cases (i.e., cases A–C), although the FSDCH model slightly overpredicts the value of 〈Σgen〉 for Δ≫δth for the higher Ret cases (i.e., cases D and E). Replacing δc=4μ0/ρ0SL (i.e., δc=2.8δz for the present thermochemistry) by δz=D0/SL in (7i)–(7iii) leads to a deterioration of the performance of the FSDCH model especially for higher Ret cases where it starts to overpredict the value of 〈Σgen〉 for Δ≫δth. However, the performance of the FSDCH model even with δz in (7i)–(7iii) remains better than the FSDA and FSDC models for all cases considered here. The FSDK model (see (8) underpredicts the value of 〈Σgen〉 for all Δ for all cases. However, the level of underprediction decreases for larger Δ.
Percentage error 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 and Δ=24Δm=2.4δth for (a–e) cases A–E.
The PEs for the FSDCH, FSDF (see (9)), and FSDNEW (see (12)) models remain negligible in comparison to the PEs for all the other models considered. Figure 3 shows that the FSDF model underpredicts 〈Σgen〉 slightly for small and moderate values of Ret (i.e., cases A–C), but the prediction of this model remains very close to the DNS result for high values of Ret (cases D and E). The PEs for the FSDCH, FSDNEW and FSDF models remain comparable. 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. Hence the FSDF model underpredicts the value of 〈Σgen〉 for small values of Δ for all the cases considered here (see Figure 3). This limitation of the FSDF model can be avoided using a modified form of (8):
(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 (5ii), 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). The model given by (14) is denoted as the MFSDF model. Figure 3 shows that the modification given by (14) does not appreciably alter the performance of (8), but this modification ensures that the model given by (8) will approach the correct asymptotic limit (i.e., limΔ→0Σgen=limΔ→0|∇c-|=|∇c|) for very small Δ (i.e., Δ→0). Note that the combination of parameterisation of D and Γ according to [19] and (5ii), 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 a power-law scheme Ξ=(η0/ηi)D-2 which is strictly valid only for Δ which are sufficiently greater than ηi (i.e., Δ≫ηi), as can be seen from Figure 2. Hence the predictive capability of the FSDK model improves when Δ>ηi (see Figure 3). However, the FSDK model underpredicts Σ_{gen} because the inner cut-off scale is taken to be 3δz in this model whereas ηi=1.785δz 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 [3], where Sd=(Dc/Dt)/|∇c|=[w˙+∇·(ρD∇c)]/ρ|∇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) as [17, 24, 29, 30] Sr=w˙/ρ|∇c|, Sn=N→·∇(ρDcN→∇c)/ρ|∇c|, and St=-Dc∇·N→. It has been shown previously [6, 7, 9] that (aT)¯s remains positive and thus acts to generate flame area, whereas (Sd∇·N→)¯s=[(Sr+Sn)∇·N→]¯s-[Dc(∇·N)2]¯s is primarily responsible for flame area destruction. The equilibrium of flame area generation and destruction yields (K)¯s=0, which leads to [8] (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 TRZ regime [17]. However, most algebraic models (e.g., FSDA, FSDC, 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 the term -[Dc(∇·N→)2]¯s was ignored [10–14]. Hence these models underestimate the flame surface area destruction in the TRZ regime, which leads to overprediction of 〈Σgen〉 for the FSDA, FSDC, and FSDW models. Although the FSDCH model was proposed for the CF regime (where the term (-[Dc(∇·N→)2]¯s) was ignored), the efficiency function was tuned to capture the expected behaviour of the turbulent flame speed ST=ΞSL for both δz<η (as in the CF regime) and δz>η (as in the TRZ regime). Hence this model is somewhat capable of predicting the behaviour of 〈Σgen〉 for the TRZ regime flames considered here. However, this model starts to overpredict due to underestimation of the destruction of flame surface area in the TRZ regime for higher Ret cases where the effects of (-[Dc(∇·N→)2]¯s) are significant. Moreover, the performance of this model is sensitive to the definition of the flame thickness used in the efficiency function.
Criterion 2.
To assess model performance with respect to Criterion 2, the variations of mean Σgen conditionally averaged on c- are shown in Figure 4 for Δ=8Δm=0.8δth and in Figure 5 for Δ=24Δm=2.4δth. These Δ values have been chosen since they correspond to Δ<ηi and Δ>ηi respectively. The following observations can be made from Figure 4.
The FSDW model tends to overpredict the value of Σgen for flames with higher Ret (e.g., cases D and E). However, the extent of this overprediction is relatively lower in the low Ret cases (e.g., cases A and B).
The FSDW model tends to predict a peak value of Σgen at c->0.6, whereas the peak value of Σgen obtained from DNS is attained close to c-≈0.6 for all the cases.
The FSDK model tends to underpredict the value of Σgen for all cases. The physical explanations provided earlier for the underprediction of 〈Σgen〉 by the FSDK model (see Figure 3) are also valid here.
Models FSDA, FSDC, FSDCH, FSDF, and FSDNEW all tend to capture the Σgen variation with c- obtained from DNS data. The prediction of the MFSDF model remains comparable to that of the FSDF model.
Comparing Figure 4 with Figure 5 reveals that there is significantly greater spread between the predictions of the various Σgen models for Δ=2.4δth than for Δ=0.8δth. The following observations can be made from Figure 5.
Models FSDW, FSDA, and FSDC all overpredict the value of Σgen, and the overprediction increases with increasing Ret. The FSDCH model captures the behaviour of Σgen for small values of Ret (e.g., cases A–C), but it overpredicts the value of Σgen for higher Ret cases (e.g., cases D–E).
Similar to Figure 4, the FSDW model predicts a peak at c-> 0.6, whereas the peak value of Σgen, from DNS occurs at c-≈0.5 for all cases.
Models FSDF, FSDK, FSDNEW and MFSDF predict Σgen satisfactorily throughout the flame brush.
The difference between the models MFSDF and FSDF seems to be very small, and both predict Σgen satisfactorily.
Variation of the mean values of Σgen×δz conditional on c- across the flame brush for Δ=8Δm=0.8δth according to the DNS () and model () for cases A (1st row), B (2nd row), C (3rd row), D (4th row), and E (5th row).
Variation of the mean values of Σgen×δz conditional on c- across the flame brush for Δ=24Δm=2.4δth according to the DNS () and model () for cases A (1st row), B (2nd row), C (3rd row), D (4th row), and E (5th row).
Unlike any of the other models, the prediction of the FSDK model improves with increasing Δ, which is consistent with observations made in the context of Figure 3. Moreover, the prediction of the FSDW model remains skewed towards the burned products due to the c~ dependence of Ξ (i.e., Ξ=1+1.24c~uΔ′/SLReη). The models FSDW, FSDA, and FSDC underestimate the destruction of flame surface area in the TRZ regime by ignoring Σgen destruction arising due to -[Dc(∇·N→)2]¯s which eventually leads to the overprediction of Σgen. As discussed above, the efficiency function Γ in the FSDCH model is parameterised to capture the turbulent flame speed behaviour in both the CF and TRZ regimes, and hence this model performs satisfactorily with respect to criterion 2 for all cases considered here.
The use of local Karlovitz number KaΔ and turbulent Reynolds number RetΔ enables local variation of D, and a satisfactory performance of the FSDNEW model with respect to Criteria 1 and 2 indicates that (11) satisfactorily captures the local Reynolds and Karlovitz number dependences of D for the present definitions of these numbers (i.e., KaΔ=6.66(k~Δ/SL)3/2(Δ/δz)-1/2 and RetΔ=4.0(ρ0uΔ′Δ/μ0)).
Criterion 3.
The correlation coefficients between the DNS result and the model prediction for Σgen are shown in Figure 6 for 0.1≤c-≤0.9. For c-<0.1 and c->0.9,Σgen is small and thus the correlation coefficients in these regions of the flame brush do not have much physical significance. On comparing cases A–E, it is clear that the correlation coefficients are significantly affected by Δ and by Ret. With increasing Ret, the correlation coefficients decrease significantly with increasing Δ which indicates that the accuracy of the models deteriorates for large Δ, as flame wrinkling increasingly takes place at the sub-grid scale with increasing Δ. A modelled transport equation [6, 7, 9, 10, 18] for Σgen may be advantageous in terms of capturing the correct strain rate and the curvature dependences of Σgen, provided the unclosed terms of Σgen transport equation are accurately modelled.
Correlation coefficients between modelled and actual values of Σgen in the c- range 0.1≤c-≤0.9 for Δ=4Δm=0.4δth; Δ=8Δm=0.8δth; Δ=12Δm=1.2δth; Δ=16Δm=1.6δth; Δ=20Δm=2.0δth and Δ=24Δm=2.4δth for (a–e) cases A–E.
It has been shown in Figures 3–6 that the FSDCH, FSDF, MFSDF, and FSDNEW models perform better than the other models in the thin reaction zones regime flames considered here. Note that the MFSDF model is now considered instead of the FSDF model since it removes the limitations of the FSDF model when the flow is fully resolved (i.e., Δ→0 and uΔ′→0). Moreover, the performance of the FSDCH, MFSDF, and FSDNEW models remains comparable, and thus any of these models is likely to provide the most satisfactory prediction within the thin reaction zones regime.
5. Conclusions
The performance of several wrinkling-factor-based algebraic models for Σgen for flames within the TRZ regime has been assessed based on DNS of turbulent premixed flames over a range of different values of Ret. It has been found that the fractal dimension D increases with increasing Ret and Ka before reaching an asymptotic value. By contrast, the inner cut-off scale ηi is not significantly affected within the range of Ret considered. The observed behaviours of D and ηi have been incorporated into a new power-law model for Σgen in the context of LES. Various criteria have been used to assess the performance of this model along with the other existing models. Most models show a deterioration of performance with increasing Δ, and in general the performance of the models is better for lower Ret. Based on this assessment, models have been identified which predict Σgen satisfactorily for all the cases considered here and for different values of Δ. It is worth noting that the present study has been carried out for a range of moderate Ret without the effects of detailed chemistry and transport. Thus, three-dimensional DNS with detailed chemistry along with experimental data for higher values of Ret will be necessary for a more comprehensive analysis.
Acknowledgment
The authors are grateful to EPSRC, UK, for financial assistance.
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