Extensive Discussions of the Eddy Dissipation Concept Constants and Numerical Simulations of the Sandia Flame D

Abstract

The indisputable wide use of the Eddy Dissipation Concept (EDC) implies that the resulting mean reaction rate is reasonably well modeled. To model turbulent combustions, an amount of EDC constants that differ from the original values was proposed. However, most of them were used without following the nature of the model or considering the effects of the modification. Starting with the energy cascade and the EDC models, the exact original primary and secondary constants are deduced in detail in this work. The mean reaction rate is then formulated from the primary constants or the secondary constants. Based on the physical meaning of fine structures, the limits of the EDC constants are presented and can be used to direct the EDC constant modifications. The effects of the secondary constant on the mean reaction rate are presented and the limiting turbulence Reynolds number used for the validity of EDC is discussed. To show the effects of the constants of the EDC model on the mean reaction rate, 20 combinations of the primary constants are used to simulate a laboratory-scale turbulent jet flame, i.e., Sandia Flame D. After a thorough and careful comparison with experiments, case 8, with a secondary constant of 6 and primary constants of 0.1357 and 0.11, can aptly reproduce this flame, except for in the over-predicted mean OH mass fraction.

1. Introduction

First developed by B.F. Magnussen [1,2], the Eddy Dissipation Concept (EDC) model has been widely used for combustion predictions. The main idea of the EDC is that chemical reactions in turbulent flows occur in intermittent fine structures or small eddies. The species gradients and molecular mixing coexist with velocity gradients such as viscous dissipation. The EDC consists of an energy cascade model and a reactor model. The cascade model links the fine structures, where reactions are assumed to occur, to the mean turbulence field resolved by the Reynolds-averaged Navier –Stokes (RANS) equation. These smaller eddies are regarded as a reactor, and the reaction rate is formulated from the species mass balance for this reactor [1,2,3,4,5,6]. The indisputable wide use of the EDC model indicates that the mean reaction rate is reasonably well modeled.

During the last decade, various modifications have been suggested to the EDC constants, i.e., the primary EDC constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ or the secondary EDC constants $C_{\mathsf{\tau }}$ and $C_{\mathsf{\gamma }}$. According to the EDC model, $C_{\mathsf{\tau }}$ is a function of $C_{\mathrm{D}2}$ while $C_{\mathsf{\gamma }}$ is a function of both $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ [5]. To the best of our knowledge, Rehm et al. [7] were the first to propose changing the constants for modeling high-pressure gasification processes, where $C_{\mathsf{\tau }}$ up to 8 and $C_{\mathsf{\gamma }}$ up to 13 were used. These selections varied significantly the original values, i.e., $C_{\mathrm{D}1}=0.135$ and $C_{\mathrm{D}2}=0.50$, while $C_{\mathsf{\tau }}=0.4082$ and $C_{\mathsf{\gamma }}=2.1298$. However, detailed effects of the modification were barely found. In addition, some misunderstandings have emerged during the use of the EDC model. For example, modified constants $C_{\mathsf{\tau }}=3.0$ and $C_{\mathsf{\gamma }}=$2.1377 [8] were tried for the numerical investigation of turbulent natural gas combustions for jets in a coflow of lean combustion products in the Delft-Jet-in-Hot-Coflow (DJHC) burner. A numerical study of a jet-in-hot coflow (JHC) burner emulating Moderate or Intense Low-oxygen Dilution (MILD) combustion conditions was performed with modified $C_{\mathsf{\tau }}=$ 1.5 or 3.0 and $C_{\mathsf{\gamma }}=1.0$ [9]. A numerical study of a reversed-flow small-scale combustor was reported with modified $C_{\mathsf{\gamma }}=2.13$ or 5.0 [10]. A natural gas non-catalytic partial oxidation reformer was simulated with modified $C_{\mathsf{\gamma }}=2.14$, 3.2, 4.4, or 5.56 [11]. Lifted jet flames in a heated and depleted oxygen coflow stream were simulated with $C_{\mathsf{\tau }}=$ 3.0 and $C_{\mathsf{\gamma }}=0.5$, 0.75, 1.0, 1.5, or 2.5 [12]. The chemical kinetic effect of the addition of hydrogen to ethylene jet flames in a hot coflow was investigated with modified $C_{\mathsf{\gamma }}=3$ and $C_{\mathsf{\tau }}=1.0$ [13]. Biomass combustion was modeled with modified $C_{\mathsf{\tau }}=$ 1.0, 1.5, or 5.62 and $C_{\mathsf{\gamma }}=1.12$, 1.75, 1.8, 2.37, or 2.46 [14,15]. The gas-phase combustion of a grate-firing biomass furnace was modeled with modified constants $C_{\mathsf{\tau }}=$ 1.0, 1.5, or 5.62 and $C_{\gamma }=1.12$, 1.75, 1.8, 2.37, or 2.46 [16]. Ethylene jet flames in diluted and heated oxidant stream combustion conditions were modeled [17] with $C_{\mathsf{\tau }}=3.0$ and $C_{\mathsf{\gamma }}=1.0$, which are the same as those used by De et al. [8]. The Adelaide Jet in Hot Co-flow (JHC) burner at different co-flow compositions and fuel-jet Reynolds numbers was modeled with modified $C_{\mathsf{\tau }}=0.82$, 1.0, 1.07, 1.25, 1.47, 1.77, or 1.96 and $C_{\mathsf{\gamma }}=1.78$, 1.9, 2.0, or 2.14 [18]. Turbulence–chemistry interactions under hot diluted combustion of CH₄/H₂ were modeled with modified $C_{\mathsf{\tau }}=0.0893$, 0.2886, 0.5773, or 3.0 and $C_{\mathsf{\gamma }}=1.0$, 1.7976, 2.5422, or 5.795 [19]. The Adelaide Jet in Hot-Coflow burner was modeled with modified constants, $C_{\mathsf{\tau }}=1.47$ and $C_{\mathsf{\gamma }}=1.5$ or 1.9 [20], which are not the same used by Parente et al. [18]. The range of validity of the values proposed for those constants is not well established. It is well-known that when constants $C_{\mathsf{\tau }}$ and $C_{\mathsf{\gamma }}$ are modified, constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ will also differ from the original values. The value of $C_{\mathrm{D}1}$ corresponds to three-half of $C_{\mathsf{\mu }}$, which was set to be the square of a value of 0.3 for the ratio of turbulent shear stress to turbulence energy [5,6]. The value of 0.3 can be discussed, but two-equation turbulence models have had undeniable success; minor departures from these data are not likely to cause significant problems. However, large modifications of $C_{\mathrm{D}1}$ will introduce inconsistency between the energy cascade model and the turbulence model. The second constant, $C_{\mathrm{D}2}$, related decaying turbulence and the ratio of kinetic energy transfer to smaller scales to the viscous dissipation [5,6]. Mardani [19] pointed out that $C_{\mathrm{D}1}$ should not be changed in order to preserve consistency with a turbulence model.

Some of the aforementioned studies used EDC constants that are empirically proposed, and some combinations lead nonphysical meanings. For example, the values of $C_{\mathsf{\tau }}=$ 1.5 or 3.0 and $C_{\mathsf{\gamma }}=1.0$ modified by Aminian et al. [9] correspond to the primary constants of $C_{\mathrm{D}1}=2.25$ and $C_{\mathrm{D}2}=6.75$, or $C_{\mathrm{D}1}=4.5$ and $C_{\mathrm{D}2}=27.0$. This shows that $C_{\mathsf{\mu }}=\frac{2}{3}{C}_{\mathrm{D}1}=1.5$ or $C_{\mathsf{\mu }}=3$ was largely modified from its original value $C_{\mathsf{\mu }}=0.09$. Ertesvåg [21] comprehensively reviewed the proposed modifications, gave an analysis of their effects, and presented three criteria for evaluating the proposed constants. These criteria should be followed when turbulence combustion is simulated.

Magnussen [6] recommended that the ratio of the fine-structure mass to the total mass should be the square of the ratio of mass in the fine-structure-containing regions to the total mass, i.e., ${\gamma }^{*}={\gamma }_{\mathsf{\lambda }}^2$. According to this recommendation, the mean reaction rate is reformulated in this work. After reviewing the energy cascade model, the original values of the primary constants, $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2},$ and the secondary constants, $C_{\mathsf{\tau }}$ and $C_{\mathsf{\gamma }},$ are obtained in this work by comparing the characteristic scales of the fine structures [1,5], which differ slightly from the commonly used values. The limits of the secondary constant for the validity of the EDC model are proposed. The mean reaction rate is formulated as the function of the EDC constants, and the effects of EDC constants on the mean reaction rate are analyzed. The limited turbulence Reynolds number is discussed for different combinations of $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, which is helpful for the correct modification of the EDC constants.

Conversely, few calculation reports of conventional flames, such as the Sandia flames, can be found in the literature using the EDC model with modified constants. Sandia flames [22,23,24,25] are a series of six piloted methane–air jet flames (Flame A to F) and have been used as a part of the Turbulent Flames (TNF) workshop (https://tnfworkshop.org/, accessed on 1 May 2022). Simultaneous measurements of major and minor species and temperatures [22], as well as detailed velocity measurements [23], are available for these flames. Since these flames have increasing velocity in their fuel and pilot jets, they have an increasing probability of local extinction in the downstream section of the flame length. Flames C and D have a very small probability of local extinction, whereas extinction becomes more prominent in flames E and F [22] due to higher jet and pilot velocity ratios.

The test case chosen for this study is the Sandia flame D, which includes a very small probability of localized extinction complying within the criteria [21], with the proposed modifications of the EDC constants. To find proper EDC constants for the Sandia Flame D, 20 combinations of $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, including those suggested by Ertesvåg [21] and those that have been used in the literature [2,5,18,19,21], have been used with the detailed chemical kinetic mechanism of GRI-Mech 3.0 [26] (consisting of 53 species and 325 elemental reactions) and the results are compared with the experiments.

Results show that numerical predictions with $C_{\mathrm{D}1}=0.1357$ and $C_{\mathrm{D}2}=0.11$ are found to be more accurate than those with original EDC constants, except for the prediction of mean OH mass fractions.

2. Reaction Rates

The approach of EDC is to solve partial differential or “transport” equations for the mean mass of individual species. Hence, the mean reaction rates have to be expressed.

The reaction rate of EDC is the result of a mass balance for the fine-structure reactor, as described by Magnussen [1,2]. The inflow has the properties of the “surroundings” (denoted by ⁰) of the fine-structure reactor and the interior and the outflow also have fine-structure properties (denoted by *), as shown in Figure 1.

The mass inflow rate divided by the mass of the fine structures is denoted by ${\dot{m}}^{*}$. The mass balance of the reactor gives a reaction rate for the reactor [21]

$$R_k^{*}={\rho }^{*}{\dot{m}}^{*}\left(Y_k^{*}-Y_k^0\right)$$

(1)

The ratio of mass in fine structures to the total mass, or the mass fraction of the fine structures, is represented by ${\gamma }^{*}$. The intermittent fine structures are assumed to be gathered in certain regions of the total flow, particularly in the interface between the bigger eddies. The ratio of mass in the fine structure regions to the total mass, known as the mass fraction of the fine structure regions, is denoted by ${\gamma }_{\mathsf{\lambda }}$. Furthermore, not all small eddies are assumed to have conditions that favor proceeding reactions. Therefore, the quantity $\chi$ is used to denote the fraction of the fine structures that react. Accordingly, ${\gamma }^{*}\chi$ is the reactor mass as a fraction of the total mass. In the original version of the EDC [1], ${\gamma }^{*}={\gamma }_{\mathsf{\lambda }}^3$.

The reactions are assumed to proceed in the fine structures. The mean reaction rate, which is the mass source term in the conservation equation for the mean species k, is expressed as [21]

$${\overline{R}}_k=\frac{\overline{\rho }}{{\rho }^{*}}{\gamma }^{*}\chi R_k^{*}$$

(2)

Substituting Equation (1) to Equation (2) yields

$${\overline{R}}_k=\frac{{\gamma }^{*}\overline{\rho }\chi }{{\tau }^{*}}\left(Y_k^{*}-Y_k^0\right)$$

(3)

where ${\tau }^{*}=\frac{1}{{\dot{m}}^{*}}$ is regarded as the characteristic time scale of the fine structures, or the fine structure residence time. This is known as the fluid dynamics time scale for chemical reactions, and is an important quantity for the treatment of chemical kinetics [3,4].

A mass-weighted average of the fine-structure reactor and its surroundings is

$${\tilde{Y}}_k={\gamma }^{*}\chi Y_k^{*}+\left(1-{\gamma }^{*}\chi \right)Y_k^0$$

(4)

then

$$Y_k^{*}-Y_k^0=\frac{Y_k^{*}-{\tilde{Y}}_k}{1-{\gamma }^{*}\chi }$$

(5)

Substituting Equation (5) to Equation (3) yields the mean reaction rate

$${\overline{R}}_k=\frac{{\gamma }^{*}\overline{\rho }\chi }{{\tau }^{*}\left(1-{\gamma }^{*}\chi \right)}\left(Y_k^{*}-{\tilde{Y}}_k\right)$$

(6)

It is convenient to define an “EDC factor” $g_{\mathrm{EDC}}$ as

$${\overline{R}}_k=g_{\mathrm{EDC}}\frac{\overline{\rho }}{{\tau }^{*}}\left(Y_k^{*}-{\tilde{Y}}_k\right)$$

(7)

For the original version [1], the EDC factor of Equation (7) is

$$g_{\mathrm{EDC}81}=\frac{{\gamma }_{\mathsf{\lambda }}^3\chi }{1-{\gamma }_{\mathsf{\lambda }}^3\chi }$$

(8)

In the 1989/1994 version [2,21], the reciprocal of ${\gamma }_{\mathsf{\lambda }}$ appeared as a factor in the reaction rate expression. Hence, the EDC factor of Equation (7) is

$$g_{\mathrm{EDC}89}=\frac{{\gamma }_{\mathsf{\lambda }}^2\chi }{1-{\gamma }_{\mathsf{\lambda }}^3\chi }$$

(9)

When using a detailed mechanism, Gran and Magnussen [4] compared the variable $\chi$ to the simple form $\chi =1$. The differences in the tested cases were modest. Ertesvåg [21] pointed out that, while $\chi$ was not a unity, it had become a part of the solution. Using $\chi =1$ as an alternative to simplify the calculations:

$$g_{\mathrm{EDC}94}=\frac{{\gamma }_{\mathsf{\lambda }}^2}{1-{\gamma }_{\mathsf{\lambda }}^3}$$

(10)

In the 2005 version [6], the fine-structure mass fraction was re-interpreted as ${\gamma }^{*}={\gamma }_{\mathsf{\lambda }}^2$, instead of ${\gamma }^{*}={\gamma }_{\mathsf{\lambda }}^3$, and the EDC factor of Equation (7) is

$$g_{\mathrm{EDC}05}=\frac{{\gamma }_{\mathsf{\lambda }}^2\chi }{1-{\gamma }_{\mathsf{\lambda }}^2\chi }$$

(11)

The EDC factor of Equation (10) is used in Ansys Fluent [28]. The following discussions are based on this equation.

3. Fine-Structures Model

The concept of the energy cascade by Richardson [29] is that the turbulence can be considered to be composed of eddies of different sizes. The large eddies are unstable and break up, which transfers their energy to somewhat smaller eddies. These smaller eddies undergo a similar break-up process, and transfer their energy to even smaller eddies. This energy cascade, in which energy is transferred to successively smaller and smaller eddies, continues until the Reynolds number is sufficiently small enough (the unity Reynolds number based on the Kolmogorov scales) that the eddy motion is stable and molecular viscosity is effective in dissipating the kinetic energy.

Magnussen [1,2] presented an energy cascade model with a characteristic frequency or strain rate. This model is used to simulate the transfer of mechanical energy from the mean flow to heat by turbulent effects, as illustrated in Figure 2, which includes infinity levels. $w^{\prime }$ represents the feed of mechanical energy from the mean flow to the turbulence. For steady-state turbulence, this is the production of turbulent kinetic energy k. The sum of q′ + q″ + … + q_n + … + q* is the dissipation rate of turbulent kinetic energy $\epsilon$. When turbulent quantities, such as k, are found from transport equations, $w^{\prime }$ represents the total supply of turbulent kinetic energy [5].

The first level in the turbulent structure is the large, energy-rich eddies that are characterized by the velocity scale u′ and length scale L′. The velocity scale u′ is related to the turbulent energy k by $k=\frac{3}{2}{u}^{\prime 2}$. A frequency or strain rate is expressed as [1,2,5]

$${\omega }^{\prime }=\frac{u^{\prime }}{L^{\prime }}$$

(12)

Likewise, the nth level was characterized by $u_n$ and $L_n$. In the smallest eddies (fine structures), $u^{*}$ and $L^{*}$ are of the same order of magnitude as the Kolmogorov scales.

For high and moderate Reynolds numbers, the dissipation is small at the upper levels. This means that for a small n, q_n is negligible compared to w_n, and w_n is approximately equal to w_n+1. Therefore, for n = 2, $u^{\prime \prime 2}$ can be approximately regarded as $\frac{1}{2}{u}^{\prime 2}$. Similarly, there are [5]

$$\{\begin{array}{c}{u}^{\prime \prime 2}\approx \frac{1}{2}{u}^{\prime 2} \\ w^{″}=C_{\mathrm{D}1}{\omega }^{\prime }\frac{3}{2}{u}^{\prime 2}=C_{\mathrm{D}1}{\omega }^{\prime }k=\frac{3}{2}{C}_{\mathrm{D}1}\frac{u^{\prime 3}}{L^{\prime }}\end{array}$$

(13)

When the Reynolds number is high, the transfer from the first level to the second level, w″, is much larger than the direct dissipation from the first level, i.e., q′. This indicates that the dissipation $\epsilon$ is approximately equal to the energy transfer w″, i.e.,

$$\epsilon =w^{\prime }=w^{″}+q^{\prime }\approx w^{″}=\frac{3}{2}{C}_{\mathrm{D}1}\frac{u^{\prime 3}}{L^{\prime }}$$

(14)

At the last level, the fine structures are [5]

$$w^{*}=\frac{3}{4}\epsilon =\frac{2}{3}{C}_{\mathrm{D}1}\frac{u^{*3}}{L^{*}}$$

(15)

and

$$q^{*}=\frac{3}{4}\epsilon =C_{\mathrm{D}2}\nu \frac{u^{*2}}{L^{*2}}$$

(16)

The two aforementioned equations give the characteristic scales for the fine structures, the length of which can be expressed as:

$$\{\begin{array}{c}{L}^{*}=\frac{2}{3}{\left(\frac{3C_{\mathrm{D}2}^3}{C_{\mathrm{D}1}^2}\right)}^{1/4}\eta \\ \eta ={\left(\frac{{\nu }^3}{\epsilon }\right)}^{1/4}\end{array}$$

(17)

and the velocity scale can be expressed as

$$\{\begin{array}{c}{u}^{*}={\left(\frac{C_{\mathrm{D}2}}{3C_{\mathrm{D}1}^2}\right)}^{1/4}{u}_{\mathsf{\eta }}\\ u_{\mathsf{\eta }}={\left(\nu \epsilon \right)}^{1/4}\end{array}$$

(18)

These scales are of the same order of magnitude as the Kolmogorov scales, i.e., the Kolmogorov length scale $\eta$ and the Kolmogorov velocity scale $u_{\mathsf{\eta }}$. The corresponding Reynolds number becomes

$$R{e}^{*}=\frac{u^{*}{L}^{*}}{\nu }=\frac{2C_{\mathrm{D}2}}{3C_{\mathrm{D}1}}$$

(19)

The Reynolds number based on the Kolmogorov scales is

$$R{e}_{\mathsf{\eta }}=\frac{u_{\mathsf{\eta }}\eta }{\nu }=1$$

(20)

This indicates that that motions on the Kolmogorov scales are strongly affected by viscosity.

Following Magnussen’s idea [1,21,30], the ratio of mass in these fine-structure-containing regions to the total mass, defined as ${\gamma }_{\mathsf{\lambda }}$, is

$$\{\begin{array}{c}{\gamma }_{\mathsf{\lambda }}=\frac{u^{*}}{u^{\prime }}={\left(\frac{3C_{\mathrm{D}2}}{4C_{\mathrm{D}1}^2}\right)}^{1/4}{\left(\frac{\nu \epsilon }{k^2}\right)}^{1/4}=C_{\mathsf{\gamma }}R{e}_{\mathrm{T}}^{-\frac{1}{4}}\\ C_{\mathsf{\gamma }}={\left(\frac{3C_{\mathrm{D}2}}{4C_{\mathrm{D}1}^2}\right)}^{1/4}={\left(C_{\mathrm{D}3}{C}_{\mathsf{\alpha }}\right)}^{1/4}\\ R{e}_{\mathrm{T}}=\frac{k^2}{\nu \epsilon }; C_{\mathsf{\alpha }}=\frac{C_{\mathrm{D}2}}{C_{\mathrm{D}1}^2}\end{array}$$

(21)

where $R{e}_{\mathrm{T}}$ is the turbulent Reynolds number, $C_{\mathsf{\alpha }}$ is the secondary EDC constant, and constant $C_{\mathrm{D}3}$ is $\frac{3}{4}$. Based on the physical meaning of ${\gamma }_{\mathsf{\lambda }}$, it is reasonable to require the ${\gamma }_{\mathsf{\lambda }}$ value to be less than unity. From there, $C_{\mathsf{\alpha }}\le \frac{1}{C_{\mathrm{D}3}}R{e}_{\mathrm{T}}=\frac{4}{3}R{e}_{\mathrm{T}}$ is obtained from Equation (21), which means that the proposed EDC constants are controlled by the lowest turbulent Reynolds number in the space. This is the lower limit of $C_{\mathsf{\alpha }}$ for the validity of the EDC model proposed in our previous work [31].

According to Ertesvåg [21], the turbulent Reynolds number at a certain ratio of the viscous term (${\epsilon }_2=C_{\mathrm{D}2}\nu {\omega }^2$) to the total dissipation can be expressed as

$$R{e}_{\mathrm{T}}=\frac{{\left(1-{\epsilon }_2/\epsilon \right)}^2}{{\epsilon }_2/\epsilon }\frac{C_{\mathrm{D}2}}{C_{\mathrm{D}1}^2}=\frac{{\left(1-{\epsilon }_2/\epsilon \right)}^2}{{\epsilon }_2/\epsilon }{C}_{\mathsf{\alpha }}$$

(22)

The meaningful solution of Equation (22) is the positive root for the ratio ${\epsilon }_2/\epsilon$, which should be less than unity.

The ratio of the fine-structure mass to the total mass is expressed as

$${\gamma }^{*}={\gamma }_{\mathsf{\lambda }}^3={\left(\frac{u^{*}}{u^{\prime }}\right)}^3=C_{\mathsf{\gamma }}^{3}R{e}_{\mathrm{T}}^{-\frac{3}{4}}$$

(23)

The mass transfer between fine structures and their surroundings, divided by the fine-structure mass, ${\dot{m}}^{*}$, and the characteristic time scale of the fine structures,${\tau }^{*}$, can be modeled as [1,21,30]

$$\{\begin{array}{c}{\dot{m}}^{*}=2\frac{u^{*}}{L^{*}}={\left(\frac{3}{C_{\mathrm{D}2}}\right)}^{1/2}{\left(\frac{\epsilon }{\nu }\right)}^{1/2}\\ {\tau }^{*}=\frac{1}{{\dot{m}}^{*}}=C_{\mathsf{\tau }}{\left(\frac{\nu }{\epsilon }\right)}^{1/2}\\ C_{\mathsf{\tau }}={\left(\frac{C_{\mathrm{D}2}}{3}\right)}^{1/2}={\left(C_{\mathrm{D}4}{C}_{\mathsf{\alpha }}\right)}^{1/2}\end{array}$$

(24)

where constant $C_{\mathrm{D}4}$ is $\frac{C_{\mathrm{D}1}^2}{3}$ and $C_{\mathsf{\tau }}$ is the secondary constant.

With the introduction of the Kolmogorov time scale ${\tau }_{\mathsf{\eta }}$, which depends on the turbulence quantities, Equation (24) becomes

$$\{\begin{array}{c}{\tau }^{*}=C_{\mathsf{\tau }}{\tau }_{\mathsf{\eta }}\\ {\tau }_{\mathsf{\eta }}=\frac{\eta }{u_{\mathsf{\eta }}}={\left(\frac{\nu }{\epsilon }\right)}^{1/2}\end{array}$$

(25)

Thus, $C_{\mathsf{\tau }}$ can be interpreted as the time–scale ratio of the fine structure to the Kolmogorov scale and should be less than unity, and the upper limit of $C_{\mathrm{D}2}$ is 3. The limit of $C_{\mathsf{\alpha }}$, i.e., $\frac{1}{C_{\mathrm{D}4}}$ or $\frac{3}{C_{\mathrm{D}1}^2}=163$, is subsequently obtained from Equation (24), which is the upper limit of $C_{\mathsf{\alpha }}$ for the validity of the EDC model proposed in our previous work [31].

By dividing the total mass, the mass transfer between fine structures and surroundings then becomes

$$\{\begin{array}{c}\dot{m}={\dot{m}}^{*}{\gamma }^{*}=\frac{C_{\mathsf{\gamma }}^3}{C_{\mathsf{\tau }}}\frac{\epsilon }{k}R{e}_{\mathrm{T}}^{-\frac{1}{4}}=\frac{C_{\mathsf{\gamma }}}{C_{\mathsf{\beta }}}\frac{\epsilon }{k}R{e}_{\mathrm{T}}^{-\frac{1}{4}}\\ C_{\mathsf{\beta }}=\frac{1}{C_{\mathrm{R}}}=\frac{C_{\mathsf{\tau }}}{C_{\mathsf{\gamma }}^2}=\frac{2}{3}{C}_{\mathrm{D}1}=C_{\mu }\end{array}$$

(26)

This quantity can be interpreted as the mean molecular mixing rate. When the EDC factor is expressed by one of Equations (9)–(11), it contains the following quantity, i.e.,

$$\frac{{\gamma }_{\mathsf{\lambda }}^2}{{\tau }^{*}}=\frac{3}{2C_{\mathrm{D}1}}\frac{\epsilon }{k}=C_{\mathrm{R}}\frac{\epsilon }{k}=\frac{C_{\mathsf{\gamma }}^2}{C_{\mathsf{\tau }}}\frac{\epsilon }{k}$$

(27)

The reaction rate for a chemical species, ${\overline{R}}_k$, is assumed to be a linear function of $\dot{m}$. Therefore, the choice of the model constants, $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, in the turbulence cascade affects the local combustion rate predictions.

When the EDC factor is expressed by Equation (9), the mean reaction rate, Equation (7), is

$${\overline{R}}_k=\frac{\overline{\rho }}{C_{\mathsf{\tau }}{C}_{\mathsf{\gamma }}\left(\frac{1}{C_{\mathsf{\gamma }}^3}-R{e}_{\mathrm{T}}^{-\frac{3}{4}}\right)}\frac{\epsilon }{k}\left(Y_k^{*}-{\tilde{Y}}_k\right)=\frac{\overline{\rho }}{C_{\mathsf{\beta }}\left(1-C_{\mathsf{\gamma }}^{3}R{e}_{\mathrm{T}}^{-\frac{3}{4}}\right)}\frac{\epsilon }{k}\left(Y_k^{*}-{\tilde{Y}}_k\right)$$

(28)

This implies that the mean reaction rate depends mainly on the turbulence and the secondary EDC constants: $C_{\mathsf{\tau }}$ and $C_{\mathsf{\gamma }}$ by the left equation or $C_{\mathsf{\beta }}$ and $C_{\mathsf{\gamma }}$ by the right equation. Recall that $C_{\mathsf{\beta }}=C_{\mathsf{\mu }}=0.09$, thus the mean reaction rate mainly depends on the turbulence and the secondary EDC constant, $C_{\mathsf{\gamma }}$ or $C_{\mathsf{\alpha }}$, by the right equation.

4. EDC Model Constants and the Effects on Reaction Rate

The constants used in the EDC model include the primary constants, $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, and the secondary constants, $C_{\mathsf{\gamma }}$, $C_{\mathsf{\tau }}$, and $C_{\mathsf{\beta }}$. By comparing the expressions of $w^{*}$ and $q^{*}$ in Equations (15) and (16) to those in [1,2], the relations between $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ can be expressed as

$$\{\begin{array}{c}\frac{3}{2}{C}_{\mathrm{D}1}=6{\zeta }^2\\ C_{\mathrm{D}2}=15{\zeta }^2\end{array}$$

(29)

where $\zeta$ is a numerical constant. Introducing $\zeta =0.18$ [1,2] yields the accurate original values of the primary constants $C_{\mathrm{D}1}=0.1296$ and $C_{\mathrm{D}2}=0.486$. The accurate secondary constants can also be obtained, i.e., $C_{\mathsf{\gamma }}=2.1584$, $C_{\mathsf{\tau }}=0.4025,$ and $C_{\mathrm{R}}=11.57$.

By comparing the expressions of $u^{*}$ and $L^{*}$ in Equations (17) and (18) to those in [1,2], the relations between $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ can be expressed as

$$\{\begin{array}{c}{\left(\frac{C_{\mathrm{D}2}}{3C_{\mathrm{D}1}^2}\right)}^{1/4}=1.74\\ \frac{2}{3}{\left(\frac{3C_{\mathrm{D}2}^3}{C_{\mathrm{D}1}^2}\right)}^{1/4}=1.43\end{array}$$

(30)

Other original values of the primary constants can then be obtained, i.e., $C_{\mathrm{D}1}=0.1357$ and $C_{\mathrm{D}2}=0.5066$, followed by the secondary constants, which become $C_{\mathsf{\gamma }}=2.1311$, $C_{\mathsf{\tau }}=0.4109$, $C_{\mathsf{\beta }}=0.0905$, and $C_{\mathrm{R}}=11.05$. In [21], these constants were manipulated in an approximate manner, resulting in $C_{\mathrm{D}1}=0.135$, $C_{\mathrm{D}2}=0.50$, $C_{\mathsf{\gamma }}=2.1377$, $C_{\mathsf{\tau }}=0.4082,$ and $C_{\mathrm{R}}=11.11$.

Based on Equation (14), the turbulence viscosity can then be expressed as

$${\nu }_{\mathrm{t}}=u^{\prime }{L}^{\prime }=\frac{3}{2}{C}_{\mathrm{D}1}\frac{u^{\prime 4}}{\epsilon }=\frac{2}{3}{C}_{\mathrm{D}1}\frac{k^2}{\epsilon }$$

(31)

where $\frac{2}{3}{C}_{\mathrm{D}1}\left(=C_{\mathsf{\beta }}\right)$ is equal to the constant $C_{\mathsf{\mu }}=0.09$ in the widely used $k$-$\epsilon$ model [32]. This is also the reason for the approximation of $C_{\mathrm{D}1}$ from 0.1296 or 0.1357 to 0.135.

Based on the work of Perot and de Bruyn Kops [33], Ertesvåg [21] concluded that the range of the constant ratio $\frac{C_{\mathrm{D}2}}{C_{\mathrm{D}1}^2}\left(=C_{\mathsf{\alpha }}\right)$ is from 6 to 50, followed by $C_{\mathrm{D}1}=0.135,$ and $C_{\mathrm{D}2}$ is from 0.11 to 0.91, where small values imply low Reynolds numbers. This $C_{\mathsf{\alpha }}$ value range might be larger, e.g., from 1 to 100. For example, an MILD combustion furnace had been successfully modelled with smaller $C_{\mathsf{\alpha }}$ values of 1 and 2 by He et al. [27]. Since combustion commonly increases small-scale turbulence and dissipation at high wavenumbers (smaller scales) [5], $C_{\mathrm{D}2}$ should be decreased.

The mean reaction rate, Equation (28), can also be expressed as

$${\overline{R}}_k=\frac{\overline{\rho }R{e}_{\mathrm{T}}^{\frac{3}{4}}}{C_{\mathsf{\mu }}\left(R{e}_{\mathrm{T}}^{\frac{3}{4}}-C_{\mathsf{\gamma }}^3\right)}\frac{\epsilon }{k}\left(Y_k^{*}-{\tilde{Y}}_k\right)$$

(32)

The effect of $C_{\mathsf{\gamma }}$ on ${\overline{R}}_k$, as shown by Equation (28) or Equation (32), is not straightforward due to the strong coupling of ${\overline{R}}_k$, the EDC constants, and the flow variables. Therefore, the effect of $C_{\mathsf{\gamma }}$ on ${\overline{R}}_k$ in Equation (32) can be found from the partial derivative of ${\overline{R}}_k$ with respect to $C_{\mathsf{\gamma }}$, i.e.,

$$\frac{\partial {\overline{R}}_k}{\partial C_{\mathsf{\gamma }}}=\frac{3C_{\mathsf{\gamma }}^2}{R{e}_{\mathrm{T}}^{\frac{3}{4}}-C_{\mathsf{\gamma }}^3}{\overline{R}}_k$$

(33)

It is reasonable to require that the mass fraction of the fine structure regions’ ${\gamma }_{\mathsf{\lambda }}$ value be less than unity. $C_{\mathsf{\gamma }}^4\le R{e}_{\mathrm{T}}$ is obtained from Equation (21) or Equation (23). This means the proposed EDC constants are constrained by the turbulent Reynolds number. In this case, the denominator of Equation (33) is positive. Therefore, Equation (33) shows the comparatively simple coupling between $C_{\mathsf{\gamma }}$ and the change rate of ${\overline{R}}_k$ and indicates that increasing ${\overline{R}}_k$ would require increasing $C_{\mathsf{\gamma }}$, and vice versa. Meanwhile, the largest proposed $C_{\mathsf{\gamma }}$ is limited to the lowest $R{e}_{\mathrm{T}}$ value in the space.

By reviewing the proposed EDC modifications and giving an analysis of the proposed EDC constants, Ertesvåg [21] presented three criteria to evaluate the proposed constants. The first is the fine-structure region fraction ${\gamma }_{\mathsf{\lambda }}$. If ${\gamma }_{\mathsf{\lambda }}$ is closed to unity for a high turbulent Reynolds number $R{e}_{\mathrm{T}}$, the set of constants can be ignored. A tentative limit can be set by limiting ${\gamma }_{\mathsf{\lambda }}$ < 0.75 for a $R{e}_{\mathrm{T}}$ above 250, for example. The second is a requirement that the ratio of the viscous term to the total dissipation, i.e., ${\epsilon }_2/\epsilon$ is less than 0.5 at $R{e}_{\mathrm{T}}$ > 50. A low-$R{e}_{\mathrm{T}}$ limit can, tentatively, be ${\epsilon }_2/\epsilon$ > 0.5 at $R{e}_{\mathrm{T}}$ < 1. The third is that the secondary EDC constant $C_{\mathrm{R}}$ cannot differ by one order of magnitude or more from the original EDC model.

Recently, He et al. [30] established four criteria that should be complied with when the EDC model is used for turbulent flame prediction: first, the primary constant should be unchanged with $C_{\mathrm{D}1}=\frac{3}{2}{C}_{\mathsf{\mu }}$; second, the secondary EDC constant should be unchanged with $C_{\mathsf{\beta }}=\frac{C_{\mathsf{\tau }}}{C_{\mathsf{\gamma }}^2}=\frac{2}{3}{C}_{\mathrm{D}1}=C_{\mathsf{\mu }}$; third, the upper limit of the secondary EDC constant, $C_{\mathsf{\alpha },\max }$, must correspond to $C_{\mathrm{D}2}=3$; finally, the lower limit of the secondary EDC constant is $C_{\mathsf{\alpha },\min }=R{e}_{\mathrm{T},\min }$.

5. Results and Discussion

Model comparison of flow and chemical kinetic mechanisms for methane–air combustion has been presented for Sandia Flame D [34] with Ansys Fluent, and the grid and boundaries are also used in this work. The Reynolds Stress Model (RSM) [35,36,37] was used for all the simulations. The chemical kinetic mechanism of GRI 3.0 [26] was used due to the incurred errors being at a low level. An EDC model with the primary constants $C_{\mathrm{D}1}=0.135$ and $C_{\mathrm{D}2}=0.50$ was used to model the turbulence–chemistry interaction for Sandia Flame D. Reasonable predictions [34] were obtained, with the peak of the maximum temperature over-predicted more than 100 K along the axis in comparison to the experiments [24], whereas the peak of the maximum temperature in the reaction shear layer was over-predicted by 500 K for Sandia Flame D with the original EDC model [38], even though the radial extent of the temperature distributions is well captured. This indicates that the mean reaction rate is over-predicted and should be modified according to Equation (33).

Based on the above discussions of EDC model constants, numerical predictions for Sandia Flame D, which has strong turbulence with Re = 22,400, are presented using the selected model constants as shown in

Table 1. Suggested and tried EDC constants and characteristic quantities.

Case	${\mathit{C}}_{D1}$	${\mathit{C}}_{D2}$	${\mathit{C}}_{\mathsf{\gamma }}$	${\mathit{C}}_{\mathsf{\tau }}$	${\mathit{C}}_{\mathbf{R}}$	${\mathit{C}}_{\mathit{\alpha }}$	${\mathit{C}}_{\mathit{\mu }}={\mathit{C}}_{\mathit{\beta }}$	${\mathit{\gamma }}_{\mathit{\lambda }}$$@\mathit{R}{\mathit{e}}_{\mathit{T}}=250$	$\mathbf{Limited}\mathit{R}{\mathit{e}}_{\mathit{T}}$	$\frac{{\mathit{L}}^{*}}{\mathit{\eta }}$	$\mathit{R}{\mathit{e}}^{*}$	$\mathit{R}{\mathit{e}}_{\mathit{T}}$$@{\mathit{\epsilon }}_2/\mathit{\epsilon }=0.10$	${\mathit{\epsilon }}_2/\mathit{\epsilon }$$@\mathit{R}{\mathit{e}}_{\mathit{T}}=50{}^{①}$	${\mathit{\epsilon }}_2/\mathit{\epsilon }$$@\mathit{R}{\mathit{e}}_{\mathit{T}}=1{}^{②}$	$\mathit{R}{\mathit{e}}_{\mathit{T}}$$@{\mathit{\gamma }}_{\mathit{\lambda }}=1$	Ref.
1	0.1350	0.5000	2.1298	0.4082	11.11	27	0.0900	0.54	21	1.42	2.47	222	0.2825	0.8264	20.6	[2,5]
2	0.1350	0.2700	1.8257	0.3000	11.11	15	0.0900	0.46	11	0.89	1.33	120	0.1930	0.7718	11.1	[21]
3	0.1296	0.4860	2.1584	0.4025	11.57	29	0.0864	0.54	22	1.42	2.50	234	0.2909	0.8306	21.7	[1,2]
4	0.1340	0.5000	2.1377	0.4082	11.19	28	0.0893	0.54	21	1.43	2.49	226	0.2848	0.8276	20.9	[5]
5	0.1340	0.2500	1.7976	0.2887	11.19	14	0.0893	0.45	10	0.85	1.24	113	0.1850	0.7655	10.4	[19]
6	0.1340	1.0000	2.5420	0.5773	11.19	56	0.0893	0.64	42	2.40	4.97	451	0.4004	0.8747	41.8
7	0.1340	27.0000	5.7950	3.0000	11.19	1504	0.0893	1.46	1128	28.39	134.33	12,180	0.8335	0.9745	1127.8
8	0.1357	0.1100	1.4549	0.1915	11.05	6	0.0905	0.37	4	0.45	0.54	48	0.0973	0.6661	4.5	[21]
9	0.1357	0.2100	1.7101	0.2646	11.05	11	0.0905	0.43	9	0.74	1.03	92	0.1607	0.7445	8.6
10	0.1357	0.3100	1.8850	0.3215	11.05	17	0.0905	0.47	13	0.99	1.52	136	0.2101	0.7842	12.6
11	0.1357	0.4100	2.0215	0.3697	11.05	22	0.0905	0.51	17	1.22	2.01	180	0.2503	0.8093	16.7
12	0.1357	0.5100	2.1349	0.4123	11.05	28	0.0905	0.54	21	1.44	2.51	224	0.2840	0.8272	20.8
13	0.1357	0.6100	2.2326	0.4509	11.05	33	0.0905	0.56	25	1.64	3.00	268	0.3128	0.8407	24.8
14	0.1357	0.7100	2.3189	0.4865	11.05	39	0.0905	0.58	29	1.84	3.49	312	0.3380	0.8514	28.9
15	0.1357	0.8100	2.3966	0.5196	11.05	44	0.0905	0.60	33	2.03	3.98	356	0.3602	0.8602	33.0
16	0.1357	0.9100	2.4674	0.5508	11.05	49	0.0905	0.62	37	2.22	4.47	400	0.3800	0.8675	37.1
17	0.6108	6.4827	1.9000	1.4700	2.46	17	0.4072	0.48	13	4.56	7.08	141	0.2145	0.7872	13.0	[18]
18	0.6638	9.3987	2.0000	1.7700	2.26	21	0.4425	0.50	16	5.78	9.44	173	0.2439	0.8057	16.0
19	0.2686	2.0172	2.1400	0.8200	5.58	28	0.1791	0.54	21	2.87	5.01	227	0.2855	0.8279	21.0
20	0.3275	3.0000	2.1400	1.0000	4.58	28	0.2184	0.54	21	3.49	6.11	227	0.2855	0.8280	21.0

^①${\epsilon }_2/\epsilon$ should be less than 0.5 at $R{e}_{\mathrm{T}}=$ 50. ^② ${\epsilon }_2/\epsilon$ should be larger than 0.5 at $R{e}_{\mathrm{T}}=$ 1.

.

Cases 1–4 are chosen since the primary model constants are generally used in the EDC model [2,5,20]. Cases 5–7 are chosen since those constants have been used for the predictions of turbulence–chemistry interactions under hot diluted combustion of CH₄/H₂ [19]. Cases 8–16 correspond to the limits of $C_{\mathrm{D}2}$ at 0.11 and 0.91 [21] with $C_{\mathrm{D}1}=0.1357$. Cases 17–20 are chosen since they have been used for the prediction of MILD combustion [18].

The secondary EDC constant, $C_{\mathsf{\alpha }}$, with a range from 6 to 50 [21,27] is shown in Table 1 for all 20 combinations of the primary EDC constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$. The $C_{\mathsf{\alpha }}$ value for case 7 is found largely outside the range when the primary constants $C_{\mathrm{D}1}=0.134$ and $C_{\mathrm{D}2}=27$ are used. ${\gamma }_{\mathsf{\lambda }}$ at $R{e}_{\mathrm{T}}$ = 250 in Table 1 is determined by Equation (21). Limited $R{e}_{\mathrm{T}}$, i.e., $R{e}_{\mathrm{T}}=C_{\mathsf{\gamma }}^4$, as shown in Table 1, is obtained by requiring the ${\gamma }_{\mathsf{\lambda }}$ value to be less than unity, which makes the denominators of Equations (32) and (33) positive. $\frac{L^{*}}{\eta }=\frac{2}{3}{\left(\frac{3C_{\mathrm{D}2}^3}{C_{\mathrm{D}1}^2}\right)}^{1/4}$ is the ratio of the fine structure length scale to the Kolmogorov scale from Equation (17). $R{e}^{*}$ is the fine-structure Reynolds number from Equation (19). Values of $R{e}_{\mathrm{T}}$ at ${\epsilon }_2/\epsilon$ = 0.10, ${\epsilon }_2/\epsilon$ at $R{e}_{\mathrm{T}}$ = 50, and $R{e}_{\mathrm{T}}$ = 1, following the criteria proposed by Ertesvåg [21], are obtained from Equation (22) and values of $R{e}_{\mathrm{T}}$ at ${\gamma }_{\mathsf{\lambda }}=1$ are obtained from Equation (21). The lower-than-unity length scale, $\frac{L^{*}}{\eta }$, shown in Table 1, means reactions have been assumed to occur in structures smaller than the Kolmogorov scales. Thus, the EDC model with the primary constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ for these cases assumes the reactions occur in very small spotty areas entitled as fine structures in the flow field while each fine reactor is contained in the Kolmogorov eddy, which may make the reaction zone kinetically control distribution in the reaction zone [19].

The results using the recommended models in our previous work [34] and the primary constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ in Table 1 are given in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, including the mean axial velocity, the mean temperature, the Reynolds normal stress, and the major and minor species. All simulations were carried out using the commercially available software ANSYS Fluent 2021R1 [28]. Some of the results are found to be comparable with those of the Large-Eddy simulation [39,40,41].

The turbulent Reynolds numbers of the computational domain have been checked for the simulations of 20 cases (excluding case 7, which exceeded the limited $R{e}_{\mathrm{T}}$, i.e., $R{e}_{\mathrm{T}}=C_{\mathsf{\gamma }}^4$, as shown in Table 1), except in the immediate vicinity of the walls. This indicates that the EDC model, from the view of the limited Reynolds number, is valid for all cases except for case 7, which violates the first criterion presented by Ertesvåg [21].

5.1. Axial Velocity Profiles

Compared to the experiments, the mean axial velocity profiles shown in Figure 3 are all reasonably predicted using 20 combinations of $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$. Unfortunately, no result completed overlaps with the experiments. One of the possible reasons might be the results from the turbulent model. Large differences among the cases occur in the zones where the mixtures burn intensely. From Figure 3a, it can be found that the predicted mean axial velocity along the axis of case 16 is the closest to the experiments, while the results of cases 17 and 18 deviate the most from the experiments. Therefore, these two cases are not recommended. Results of other cases stay between those of cases 8–16. Large deviations for the radial mean axial velocity profiles at x/d = 30 near the axis are found in Figure 3d. The predictions of the radial mean axial velocity for 20 cases are very close to each other, indicating that the EDC model has a negligible effect on the flow when the reactions are not very intense, as shown in Figure 7b,c,f.

5.2. Reynolds Normal Stress Profiles

Compared to the experiments, the UU Reynolds normal stress profiles shown in Figure 4 are reasonably predicted for all cases. Some over-predicted values can be found in the zones where the mixtures burn intensely. Relatively small differences in the predicted Reynolds normal stress profiles are found among the 20 cases.

The axial profiles of axial velocity and Reynolds normal stress in the cases with smaller $C_{\mathsf{\alpha }}$ values, shown in Figure 4, agree better with the experiments than the results (Figure 11 in [40]) by LES for flame D. Lysenko et al. [40] concluded that the challenges relate more to flow (turbulence) modeling than to the chemistry and combustion models and believed that discrepancies between the LES predictions and experimental data are the lack of the grid resolution and the insufficient inlet boundary conditions. In the downstream region of the higher reaction thickness zone, the centerline axial velocity magnitude of the flames is under-predicted, which can be attributed to the increased ratios of the jet as well as pilot velocities, as pointed out by Saini et al. [38].

5.3. Mean Temperature Profiles

The mean temperature profiles of the 20 combinations of the primary EDC constants are shown in Figure 5. Compared to the experiments, large differences in the predicted mean temperatures among the cases occur at the peak temperature zone where the mixtures burn intensely. One reason for the over-predicted mean temperatures is that the mean reaction rate, as shown by Equation (28) or (32), is over-predicted and should be modified according to Equation (33). Reducing the mean reaction rate requires a decrease in $C_{\mathsf{\gamma }}$, as shown by Equation (33).

From Figure 5a, the mean temperatures in cases 8, 17, and 18 along the axis are the closest to the experiments, which indicates that the mean reaction rates are reasonably predicted. Unfortunately, the EDC constant combinations of cases 17 and 18 violate the criteria presented by Ertesvåg [21] and should be excluded.

With a very large value of $C_{\mathsf{\tau }}$, the temperature predicted by case 7 deviates the most from the experiments and thus is also not recommended. In addition, the proposed constants of case 7, as shown in Table 1, violate the criteria presented by Ertesvåg [21]. Results of other cases are between those of cases 8–16. Small deviations for the radial mean temperature profiles at x/d = 7.5, 15, and 30 between the axis and the location of peak temperature can be found in Figure 5. At the locations downstream and away from the peak temperature, over-predicted values are found in all cases.

Using the original values of $C_{\mathsf{\gamma }}=2.1377$ and $C_{\mathsf{\tau }}=0.4082$, i.e., case 4 in Figure 5, recent studies [34,38] show that there are notable differences in temperature in the reacting zone. Compared to the experiments, the temperature predictions of the reacting zone have significantly improved. In most cases, the predicted temperature distributions downstream of the jet were higher than the measured data.

Instead of using the global modification of the EDC constants, as used in this work, Mardani and Nazari [42] presented the results (at x/d = 15 and 30, Fig_S5 in [42]) using the local modification of EDC constants for flame D, where the peaks are successfully reduced. However, the over-predicted mean temperature at a radius larger than that of the peak temperature is moderately improved. Conversely, one criterion [21,31] for the validity of the EDC model, i.e., $C_{\mathsf{\beta }}=\frac{1}{C_{\mathrm{R}}}=\frac{C_{\mathsf{\tau }}}{C_{\mathsf{\gamma }}^2}=\frac{2}{3}{C}_{\mathrm{D}1}=C_{\mathsf{\mu }}$, is violated because $C_{\mathsf{\beta }}=2.367C_{\mathsf{\mu }}$ from the expressions of $C_{\mathsf{\tau }}$ and $C_{\mathsf{\gamma }}$ was used in [42] with the suggested constant. From this point of view, the results obtained by the cases that comply with the criteria [21,31], such as case 8, are now more credible because they are based on the theoretically correct EDC model.

5.4. Major Species Profiles

Compared to the experiments, the mean mass fractions of major species, including CH₄, CO₂, H₂O, N₂, and O₂, are reasonably predicted by all cases, as shown in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10.

Some under-predicted mean CH₄ mass fractions along the axis are found among all the cases, as shown in Figure 6a, and occur in the zones where the mixtures burn intensely, which means that the mean reaction rates are largely over-predicted. For the radial mean CH₄ mass fraction, the results of case 16 are found to agree well with the experiments, as shown in Figure 6 at x/d = 7.5 and x/d = 15. Similarly, the results of case 8 are also acceptable. Compared to the experiments, relatively large differences in the predicted radial mean CH₄ mass fraction occur close to the axis where the mixtures burn intensely, as shown in Figure 6d. The axial profiles of the mean mass fractions of CH₄ predicted by case 8, as shown in Figure 6, agree well with the experiments and are comparable to the results (Figure 12 in [40]) by LES/EDC for flame D. The radial profiles of the mean mass fractions of CH₄ at x/d = 15 predicted by cases 8 and 9 agree better with the experiments han the results (Figure 5 in [41]) of LES/transported PDF (probability density function) for flame D.

For CO₂, some over-predicted values along the axis are found in all cases, as shown in Figure 7a, and occur in the zones where the mixtures burn intensely. Cases 8, 17, and 18 aptly predict the experiments. For the radial mean CO₂ mass fraction, the results are found under-predicted upstream of the peak and over-predicted downstream of the peak, as shown in Figure 7 at x/d = 7.5, 15, and 30. Downstream of the peak temperature, around x/d = 45, the results are slightly over-predicted, as shown in Figure 7 at x/d = 45 and 60. In the zones where the reactions of mixtures are almost done, such as the location at x/d = 60, the predicted results vary slightly among the 20 cases. Compared to the predicted results of other cases, cses 8, 17, and 18 are found to agree well with the experiments for both the axial and radial mean CO₂ mass fraction.

The predicted peaks of the mean mass fractions of CO₂ are decreased with the modified EDC constants and the largest is attributed to case 8, as expected. The axial profiles of the mean mass fractions of CO₂, as shown in Figure 7, agree better with the experiments than the LES/EDC results (Figure 12 in [40]) for flame D. The results (at x/d = 15 and 30, Fig_S5 in [42]) from local modification of EDC constants for flame D show that the peaks are successfully reduced. However, the over-predicted mass fraction of CO₂ at a radius larger than that of the actual peak of mass fraction of CO₂ is improved slightly.

Similarly, over-predicted mean H₂O mass fractions along the axis upstream of the peak and under-predicted profiles downstream of the peak are found in all cases, as shown in Figure 8a, and occur in the zones where the mixtures burn intensely. This means that the mean reaction rates are largely over-predicted. The predicted results of cases 8, 17, and 18 agree reasonably well with the experiments, which indicates that the mean reaction rates are reasonably predicted. For the radial mean H₂O mass fraction, the results are found to agree quite well upstream of the peak, and are over-predicted downstream of the peak, as shown in Figure 8 for x/d = 7.5 and 15. At the location of peak temperature, around x/d = 45, the results are slightly over-predicted, as shown in Figure 8 at x/d = 45. In the zones where the reactions of the mixtures are almost done, such as at x/d = 60, the predicted results differ within a small range among the 20 cases. Compared to the predicted results of other cases, cases 8, 17, and 18 are found to agree well with the experiments for both the axial and the radial mean H₂O mass fractions.

The predicted peaks of the mean mass fractions of H₂O are decreased with the modified EDC constants and the largest is attributed to case 8, as expected. The results (at x/d = 15 and 30, Fig_S5 in [42]) from local modification of EDC constants for flame D show that the peaks are successfully reduced. However, the over-predicted mass fraction of H₂O at a radius larger than that of the peak of mass fraction of H₂O is improved slightly.

Over-predicted mean N₂ mass fraction profiles along the axis and the radii are found in all the cases, as shown in Figure 9, and occur in the zones where the mixtures burn intensely. Very small differences are found in the predicted mean N₂ mass fraction profiles among the 20 cases.

The axial profiles of the mean mass fractions of N₂ predicted by case 8, as shown in Figure 9, agree with the experiments better than the LES/EDC results (Figure 12 in [40]) for flame D.

Under-predicted mean O₂ mass fraction profiles along the axis and the radii are found in all cases, as shown in Figure 10, and occur at the zones where the mixtures burn intensely, which indicates that the mean reaction rates are largely over-predicted. Very small differences are found for the predicted mean O₂ mass fraction profiles among the 20 cases for the zones where the reactions of mixtures are near completion, such as at x/d = 45 and 60.

5.5. Minor Species Profiles

The mean mass fraction profiles for minor species, i.e., CO, H₂, NO, and OH, using 20 combinations of the primary EDC constants C_D1 and C_D2 are shown in Figure 11, Figure 12, Figure 13 and Figure 14. Compared to the experiments, large differences in the predicted mean mass fractions of minor species among the cases and the experiments occur at the peak temperature zone where the mixtures burn most intensely. The radial peaks in minor species profiles were not accurately reproduced. Aminian et al. [43] claimed that the productions of minor species, such as CO and OH, were strongly sensitive to temperature fluctuations, especially upstream of the flame.

In Figure 11a, the mean CO mass fractions along the axis of cases 17 and 18 are found to agree well with the experiments at locations upstream of the peak temperature, but are slightly under-predicted downstream of the peak temperature. Among the cases using the primary constant $C_{\mathrm{D}2}$ in the limits, i.e., from 0.11 to 0.91, the remarkably improved predictions of case 8 agree better with the experiments than those of the other cases. Results of case 7 deviate the most from the experiments, thus this case is not recommended. In addition, the proposed constants in case 7, as shown in Table 1, violate the criteria presented by Ertesvåg [21]. Results of other cases stay between those of cases 8–16.

The results for the radial mean CO mass fraction are over-predicted at the zones upstream of the peak temperature, as shown in Figure 11. Large differences in the deviations from the experiments occur in the zones where the mixtures burn intensely, as shown in Figure 11, which may also be due to the over-predicted mean reaction rates.

The mean H₂ mass fractions, which are quite similar to the profiles of the mean CO mass fraction shown in Figure 11, are shown in Figure 12. Among the cases with the primary constant $C_{\mathrm{D}2}$ in the limits, the predicted results of case 8 agree better with the experiments than those of the other cases.

In Figure 13, the mean NO mass fractions along the axis and the radii of cases 8, 17, and 18 are found to agree well with the experiments. However, the EDC constants of cases 17 and 18 violate the criteria for the proposed EDC constants [21]. The results of the other cases are significantly over-predicted, especially at locations around the peak temperature where the mixtures burn intensely, and large deviations among the cases are found for the mean NO mass fraction profiles. The results [38] of Sandia Flame D with the original EDC constants, which is case 4 in this work, also show that the results of the mean NO mass fractions are significantly over-predicted. Therefore, the EDC constant combinations of case 8 are suggested for the modeling of Sandia flame D.

The prediction of OH is particularly challenging due to the strong nonlinearity of the species’ evolution [44]. Compared to the respected measured data, the level of agreement for OH was subpar, with largely over-predicted peaks using all of the EDC constants. All results were found to significantly over-predict around the peak value locations where the mixtures burn intensely, as shown in Figure 14. However, the results of the peak value locations and the varied tendency are correctly predicted. The calculated maximum mass fractions are significantly over-predicted in comparison to the measured data. Large deviations among the cases are found for the mean OH mass fractions. Among the cases with the primary constant $C_{\mathrm{D}2}$ in the limits, the result of case 16 is the upper profile limit, and case 8 is the lower profile limit. Results of other cases are within the limits of cases 8 and 16 except for cases 7 and 17–20.

The predictions of cases 17–20 are found to be reasonable for some quantities, such as the mean temperature and the mean major and minor mass fractions, in comparison to the experiments. However, all four cases violate the third criterion presented by Ertesvåg [21], i.e., that the secondary EDC constant, $C_{\mathrm{R}}$, should not differ by one order of magnitude or more from the original model, or that $C_{\mathsf{\beta }}$ should be kept at constant around $C_{\mathsf{\mu }}$. Meanwhile, a larger primary constant $C_{\mathrm{D}1}$, which results in a large deviation of $C_{\mathsf{\mu }}$ in the turbulent model, is used in these four cases.

By checking the secondary constant $C_{\mathsf{\tau }}$, shown in Table 1, four cases, i.e., 7, 17, 18, and 20, make $C_{\mathsf{\tau }}\ge 1$, which violates the third criterion suggested by He et al. [30]. However, $C_{\mathsf{\tau }}$, as shown in Equation (25), can be interpreted as the time–scale ratio of the fine structure to the Kolmogorov scale. The Kolmogorov time scale, ${\tau }_{\mathsf{\eta }}={\left(\frac{\nu }{\epsilon }\right)}^{1/2}$, is the survival time or the lifespan of the smallest eddy, which is supposed as the fine-structure reactor in the EDC model. Based on this assumption, $C_{\mathsf{\tau }}$ should be less than or equal to unity. Values of $\frac{2}{3}{C}_{\mathrm{D}1}$ for cases 17, 18, and 20 should be equal to the constant $C_{\mathsf{\mu }}=0.09$ in the widely used $k$-$\epsilon$ model [32] instead of deviating significantly. Therefore, cases 7, 17, 18, and 20 should be excluded.

Compared to the other cases, the remarkably improved predictions of case 8 significantly agree with the experiments, and some predictions of case 8, as previously mentioned, are found to agree better with the experiments than those by the LES predictions [39,40,41]. However, deviations are also found between the predictions and the experiments for the turbulent methane–air combustion of the Sandia flame D. There are some possible reasons that can lead to the deviations, including simplifications of the boundary settings, the turbulence, the turbulence–reaction interaction (the EDC model with globally or locally modified constants), and the radiation models. We expect to obtain better results using the advanced high-fidelity models, e.g., the full-spectrum k-distribution method [45,46,47,48,49,50,51] for radiation modelling, rather than using the weighted sum of gray gas (WSGG) model [52] used in this work.

6. Conclusions

The indisputable wide use of EDC indicates that the mean reaction rate is reasonably modeled. The fine-structure reactor model and the energy cascade model are outlined and summarized in this work in detail. The limited Reynolds number for the validity of EDC is discussed. The dependence of the mean reaction rates on the primary EDC constants, $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, and the secondary constants, $C_{\mathsf{\gamma }}$ and $C_{\mathsf{\tau }}$, are presented. The combinations of the primary EDC constants, i.e., $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, are validated using detailed experimental data from Sandia Flame D. Results of case 8 using the EDC constants of $C_{\mathrm{D}1}=0.1357$, $C_{\mathrm{D}2}=0.11$, $C_{\mathsf{\gamma }}=1.4549,$ and $C_{\mathsf{\tau }}=0.1915$ show promising improvement with respect to the original EDC constants. Summary conclusions may be drawn as follows:

(1)

The fine-structure reactor model and the energy cascade model are outlined and summarized. From the original work of Magnussen [1,2], the original values of the primary EDC constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$ and the secondary EDC constants $C_{\mathsf{\gamma }}$ and $C_{\mathsf{\tau }}$ are presented.

(2)

The limiting Reynolds number for the validity of EDC is concluded. Thus, the turbulent Reynolds number should be larger than $C_{\mathsf{\gamma }}^4$ with different combinations of the primary EDC constants $C_{\mathrm{D}1}$ and $C_{\mathrm{D}2}$, or the EDC constants need to be adjusted according to the criteria.

(3)

The secondary constant, $C_{\mathsf{\tau }}$, should be less than unity with the proposed EDC constant combinations because it is the time–scale ratio of the fine structure to the Kolmogorov scale. The upper limit of $C_{\mathsf{\alpha }}$ is determined from the unity $C_{\mathsf{\tau }}$.

(4)

The mean reaction rate is expressed as a function of the secondary constants, and its dependence on the EDC constant is presented. The dependence shows that decreasing the mean reaction rate requires deceases in $C_{\mathsf{\gamma }}$ or $C_{\mathsf{\alpha }}$, and vice versa.

(5)

Comprehensive comparison of the predictions of axial velocity, Reynolds stress, temperature, and major and minor species with experimental data shows that case 8, with the secondary constant $C_{\mathsf{\alpha }}=6$ or the primary constants $C_{\mathrm{D}1}=0.1357$ and $C_{\mathrm{D}2}=0.11,$ can properly reproduce the Sandia flames D.