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Journal of Combustion
Volume 2012 (2012), Article ID 794671, 17 pages
http://dx.doi.org/10.1155/2012/794671
Research Article

A Priori Assessment of Algebraic Flame Surface Density Models in the Context of Large Eddy Simulation for Nonunity Lewis Number Flames in the Thin Reaction Zones Regime

1School of Mechanical and Systems Engineering, Newcastle University, Claremont Road, Newcastle upon Tyne NE1 7RU, UK
2Engineering Department, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK

Received 21 March 2012; Accepted 24 June 2012

Academic Editor: Andrei N. Lipatnikov

Copyright © 2012 Mohit Katragadda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [111]. In the context of LES the generalised FSD (Σgen) is defined as follows [311]: Σgen=||||.𝑐(1) 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=(𝑌𝑅0𝑌𝑅)/(𝑌𝑅0𝑌𝑅) 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 [1216], which is often used in the context of thickened flame modelling [13, 14]. The wrinkling factor Ξ is closely related to Σgen according to [1216]:ΣΞ=gen||𝑐||.(2a)

Often, Ξ is expressed in terms of a power-law expression [7, 9, 13, 14] Ξ=(𝜂0/𝜂𝑖)𝐷2 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: Σgen||=Ξ𝑐||=Δ𝜂𝑖𝐷2||𝑐||,(2b)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=𝑆3𝐿/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 H2 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, 2428] 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., [111]) 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.(i)To understand the effects of Lewis number on 𝐷 and 𝜂𝑖 in the context of LES modelling.(ii)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.(iii) 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:Σgen=𝑢1+𝑎ΓΔ𝑆𝐿||𝑐||,(3a) where 𝑎=1.0 is a model parameter, 𝑢Δ=2̃𝑘Δ/3 is the subgrid turbulent velocity fluctuation, ̃𝑘Δ=(𝑢𝑖𝑢𝑖̃𝑢𝑖̃𝑢𝑖)/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: 𝑢Γ=0.75exp1.2Δ𝑆𝐿0.3Δ𝛿𝑧2/3(3b)Weller et al. [12] also presented an algebraic model for Ξ, which can be recast in the form (FSDW model): Σgen=[]||1+2̃𝑐(Θ1)𝑐||,(4) where Θ=1+0.62𝑢Δ/𝑆𝐿Re𝜂 and Re𝜂=𝑢Δ𝜂/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: Σgen=𝑢1+𝛼ΓΔ𝑆𝐿||𝑐||,(5) where Γ is given by (3b), 𝛼=𝛽×2ln(2)/[3𝑐𝑚𝑠(Re𝑡1/21)] with Re𝑡=𝜌0𝑢𝑙/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): Σgen=Δ1+min𝛿𝑧,ΓΔ𝑢Δ𝑆𝐿𝛽1||𝑐||,(6) with the efficiency functionΓΔ=(𝑓𝑎1𝑢+𝑓𝑎1Δ)1/𝑎1𝑏1+𝑓𝑏1Re1/𝑏1,(7a) where ReΔ=𝑢ΔΔ/𝜈 and with model constants 𝑏1=1.4, 𝛽1=0.5, 𝐶𝑘=1.5, and functions 𝑎1, 𝑓𝑢, 𝑓Δ, and 𝑓Re are defined by: 𝑎1𝑢=0.60+0.20exp0.1Δ𝑆𝐿Δ0.20exp0.01𝛿𝑧,𝑓𝑢=427𝐶110𝑘1/218𝐶55𝑘𝑢Δ𝑆𝐿2,𝑓Δ=27𝐶110𝑘𝜋4/3Δ𝛿𝑧4/311/2,𝑓Re=955exp(1.5𝐶𝑘𝜋4/3ReΔ1)1/2ReΔ1/2.(7b) Knikker et al. [7] proposed a model for Σgen (FSDK model) as: Σgen=Δ𝜂𝑖𝛽𝑘||𝑐||,(8) 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: Σgen=Γ𝑢Δ𝑆𝐿𝐷2||𝑐||,(9) 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=(𝑇ad𝑇0)/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=(𝑇ad𝑇0)/𝑇0, Damköhler number Da=𝑙𝑆𝐿/𝑢𝛿th, Karlovitz number Ka=(𝑢/𝑆𝐿)3/2(𝑙𝑆𝐿/𝛼𝑇0)1/2 and turbulent Reynolds number Re𝑡=𝜌0𝑢𝑙/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(𝑇ad𝑇0)/𝑇2ad=6.0), where 𝑇ac is the activation temperature.

tab1
Table 1

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, 810, 14, 3437] 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(𝑥𝑟)𝐺(𝑟)𝑑𝑟 using a Gaussian kernel given by the expression 𝐺(𝑟)=(6/𝜋Δ2)3/2exp(6𝑟𝑟/Δ2). The results will be presented for Δ ranging from Δ=4Δ𝑚0.4𝛿th to Δ=24Δ𝑚2.4𝛿th, where Δ𝑚 is the DNS grid spacing (Δ𝑚0.1𝛿th). These filter sizes are comparable to the range of Δ used in a priori DNS analysis in several previous studies [3, 810, 14], and span a useful range of length scales from Δ comparable to 0.4𝛿th0.8𝛿𝑧, where the flame is partially resolved, up to 2.4𝛿th4.8𝛿𝑧, 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=0.4𝛿th0.8𝛿𝑧) to well within the TRZ regime (for Δ0.5𝛿th𝛿𝑧) 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 Discussion

4.1. Effects of Le on 𝐷 and 𝜂𝑖

The power law expression (2b) for Σgen may be rewritten as: Σloggen||𝑐||𝜂=(𝐷2)logΔ(𝐷2)log𝑖,(10) 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 Δ𝛿𝑧 but becomes nonlinear for Δ𝛿𝑧. 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/𝛿th1.0), which is about twice the Zel’dovich flame thickness 𝛿𝑧 for the present thermochemistry (i.e., 𝜂𝑖=1.79𝛿𝑧𝛿th). 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.

fig1
Figure 1: Variation of Σgen/𝑐 with Δ/𝛿𝑧 on a log-log plot for (a–e) cases A–E. Prediction of Σgen/|𝑐|=(Δ/𝜂𝑖)𝐷2 with 𝜂𝑖 obtained from DNS and (𝐷2) according to (11) is also shown.

Contours of reaction progress variable 𝑐 in the 𝑥1𝑥2 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, 2428]. 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/𝑆𝐿Re𝑡1/4Ka1/2, 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 Le1.0 (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: 1𝐷=2+3erf(3.0Ka)1exp0.1Re𝑡𝐴𝑚1.6Le0.45,(11) where 𝐴𝑚7.5 is a model parameter. Further details on the basis of this parameterisation are given in Appendix A.

fig2
Figure 2: Contours of 𝑐 in the 𝑥1𝑥2 midplane at time 𝑡=𝛿th/𝑆𝐿 for (a–e) cases A–E.

The prediction of Σgen/|𝑐|=(Δ/𝜂𝑖)𝐷2 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=2̃𝑘Δ/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(𝜌0𝑢ΔΔ/𝜇0). The choice of model constants 𝐶Ka=6.6 and 𝐶Re=4.0 ensures an accurate prediction of 𝐷 for Δ𝜂𝑖 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): Σgen=||𝑐||Δ(1𝑓)+𝑓𝜂𝑖𝐷2,(12) where 𝑓 is a bridging function which increases monotonically from zero for small Δ (i.e., Δ/𝛿th0 or Δ𝛿th) to unity for large Δ (i.e., Δ𝜂𝑖 or Δ𝛿th). Equation (12) ensures that Σgen approaches |𝑐|(Δ/𝜂𝑖)𝐷2 for large Δ and at the same time Σgen approaches |𝑐| (i.e., limΔ0Σgen=limΔ0|𝑐|=|𝑐|) for small Δ. It has been found that Σgen|𝑐| provides better agreement with Σgen obtained from DNS data for Δ0.8𝜂𝑖, whereas the power-law Σgen=|𝑐|(Δ/𝜂𝑖)𝐷2 starts to predict Σgen more accurately for Δ1.2𝜂𝑖 (see Figure 1). Based on this observation, the bridging function 𝑓 is taken to be 𝑓=1/[1+exp{60(Δ/𝜂𝑖1.0)}], which ensures a smooth transition between 0.8𝜂𝑖<Δ<1.2𝜂𝑖. As 𝜂𝑖 is found to scale with 𝛿𝑧 (i.e., 𝜂𝑖1.79𝛿𝑧𝛿th 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 Σgen

The inaccuracy in the model predictions of Σgen can be characterised using a percentage error (PE): ΣPE=genmodelΣgenΣgen×100,(13) where Σgenmodel 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 Δ𝛿th, especially for cases with Le0.6 (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.

fig3
Figure 3: 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., Δ𝛿𝑧) but they overpredict Σgen for large values of Δ (i.e., Δ𝛿𝑧). The FSDF model (9) predicts accurately for small Δ, and marginally underpredicts for larger Δ, for cases with Le0.6. 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 Le1.0 (i.e., cases C–E), although the FSDCH model overpredicts Σgen for Δ𝛿th for cases with Le1 (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., lim𝑢Δ0Σgen=lim𝑢Δ0|𝑐|=|𝑐|) when 𝑢Δ vanishes because the flow tends to be fully resolved (i.e., limΔ0𝑢Δ=0 and limΔ0Σgen=|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): Σgen=||𝑐||Γ𝑢(1𝑓)+𝑓Δ𝑆𝐿𝐷2,(14) where 𝑓=1/[1+exp{60(Δ/𝛿th1.0)}] 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., Δ𝜂𝑖 or Δ0), where 𝑢Δ also vanishes (i.e., limΔ0𝑢Δ=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)𝐷2 which is strictly valid only for filter sizes Δ which are sufficiently greater than 𝜂𝑖 (i.e., Δ𝜂𝑖), 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 𝜂𝑖1.79𝛿𝑧 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.,  Δ𝜂𝑖). Moreover, Σgen vanishes when Δ0 according to the FSDK model, whereas Σgen should approach |𝑐| when Δ0 (i.e. lim𝑢Δ0Σgen=lim𝑢Δ0|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+𝑆𝑑𝑁 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=(𝛿𝑖𝑗𝑁𝑖𝑁𝑗)𝜕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) [810, 40, 41]: 𝑆𝑟=̇𝑤𝜌||||𝑐,𝑆𝑛=𝑁𝜌𝐷𝑐𝑁𝑐𝜌||||,𝑆𝑐𝑡=𝐷𝑐𝑁.(15) 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 (𝑆𝑑𝑁)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]: 𝑎𝑇𝑠=𝑆𝑟+𝑆𝑛𝑁𝑠+[𝐷𝑐𝑁)2𝑠.(16)

The stretch rate induced by [𝐷𝑐(𝑁)2]𝑠 becomes the leading order sink term in the thin reaction zones regime [810, 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, 1214]. 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 Σgen

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.(i)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.(ii)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.(iii)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.

fig4
Figure 4: 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.
fig5
Figure 5: 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.(i)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.(ii)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.(iii)The models FSDF, FSDK, FSDNEW, and MFSDF tend to predict the conditionally averaged value of Σgen satisfactorily throughout the flame brush.(iv)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 (PE2): PE2=ΣMODELcondΣDNScondΣmaxcond×100,(17) where ΣMODELcond and ΣDNScond are the mean values of Σgen conditional on 𝑐 as obtained from model prediction and DNS respectively, and Σmaxcond 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 PE2 for the FSDW model in the Le = 1.2 case is 15.2%, and the variation of PE2 with 𝑐 in this case is qualitatively similar to the Le = 1.0 case considered here).

fig6
Figure 6: 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.
fig7
Figure 7: 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 47, 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+1.24̃𝑐𝑢Δ/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 Σgen 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 0.1𝑐0.9 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].

fig8
Figure 8: Correlation coefficients between the modelled and the actual values of Σgen in the 𝑐 range 0.1𝑐0.9 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., 𝜂𝑖𝛿th). 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.

Appendix

A. Effects of Re𝑡 on Fractal Dimension 𝐷

The effects of Re𝑡 on 𝐷 have been analysed based on a simplified chemistry based DNS database [43, 44], in which the variation of Re𝑡Da2Ka2 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=(𝑇ad𝑇0)/𝑇0, Damköhler number Da=𝑙𝑆𝐿/𝑢𝛿th, Karlovitz number Ka=(𝑢/𝑆𝐿)3/2(𝑙𝑆𝐿/𝛼𝑇0)1/2, and turbulent Reynolds number Re𝑡=𝜌0𝑢𝑙/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)𝐷2 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.

fig9
Figure 9: 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)𝐷2 with 𝜂𝑖 obtained from DNS and (𝐷2) according to (11) is also shown.

Acknowledgment

The authors are grateful to EPSRC, UK, for financial assistance.

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