## 1 Introduction
It is well-known that when constants \(C_{\tau}\) and \(C_{\gamma}\) are modified, constants \(C_{\text{D1}}\) and \(C_{\text{D2}}\) will also differ from the original values. The value of \(C_{\text{D1}}\) corresponds to three-half of \(C_{\mu}\), 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 \(C_{\text{D1}}\) will introduce inconsistency between the energy cascade model and the turbulence model. The second constant, \(C_{\text{D2}}\), related decaying turbulence and the ratio of kinetic energy transfer to smaller scales to the viscous dissipation [5,6]. [PERSON] [19] pointed out that \(C_{\text{D1}}\) 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 \(C_{\tau}=1.5\) or 3.0 and \(C_{\gamma}=1.0\) modified by [PERSON] et al. [9] correspond to the primary constants of \(C_{\text{D1}}=2.25\) and \(C_{\text{D2}}=6.75\), or \(C_{\text{D1}}=4.5\) and \(C_{\text{D2}}=27.0\). This shows that \(C_{\mu}=\frac{2}{3}C_{\text{D1}}=1.5\) or \(C_{\mu}=3\) was largely modified from its original value \(C_{\mu}=0.09\). [PERSON] [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. [PERSON] [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., \(\gamma^{*}=\gamma_{\lambda}^{2}\). 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, \(C_{\text{D1}}\) and \(C_{\text{D2}}\), and the secondary constants, \(C_{\tau}\) and \(C_{\gamma}\), 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 \(C_{\text{D1}}\) and \(C_{\text{D2}}\), which is helpful for the correct modification of the EDC Tconstants. 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/](https://tnfworkshop.org/)]([https://tnfworkshop.org/](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.