The cross-scalar dissipation rate of reaction progress variable and mixture fraction \(\widetilde{{\varepsilon_{c\xi } }}\) plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of \(\widetilde{{\varepsilon_{c\xi } }}\) have been analysed using a direct numerical simulation (DNS) database of statistically planar turbulent stratified flames with a globally stochiometric mixture. A parametric analysis has been conducted by considering a number of DNS cases with a varying initial root-mean-square velocity fluctuation \(u^{\prime }\) and initial scalar integral length scale \(\ell_{\phi }\). The explicitly Reynolds averaged DNS data suggests that the linear relaxation model for \(\widetilde{{\varepsilon_{c\xi } }}\) is inadequate for most cases, but its performance appears to improve with increasing initial \(\ell_{\phi }\) and \(u^{\prime }\) values. An exact transport equation for \(\widetilde{{\varepsilon_{c\xi } }}\) has been derived from the first principle, and the budget of the unclosed terms of the \(\widetilde{{\varepsilon_{c\xi } }}\) transport equation has been analysed in detail. It has been found that the terms arising from the density variation, scalar-turbulence interaction, chemical reaction rate and molecular dissipation rate play leading order roles in the \(\widetilde{{\varepsilon_{c\xi } }}\) transport. These observations have been justified by a scaling analysis, which has been utilised to identify the dominant components of the leading order terms to aid model development for the unclosed terms of the \(\widetilde{{\varepsilon_{c\xi } }}\) transport equation. The performances of newly proposed models for the unclosed terms have been assessed with respect to the corresponding terms extracted from DNS data, and the newly proposed closures yield satisfactory predictions of the unclosed terms in the \(\widetilde{{\varepsilon_{c\xi } }}\) transport equation.

反应进程变量和混合分数\(\widetilde{{\varepsilon_{c\xi } }}\)的交叉标量耗散率在分层燃烧建模中起着重要作用。\(\widetilde{{\varepsilon_{c\xi } }}\)的演变和统计行为已使用具有全局化学计量混合物的统计平面湍流分层火焰的直接数值模拟 (DNS) 数据库进行了分析。通过考虑许多具有不同初始均方根速度波动\(u^{\prime }\)和初始标量积分长度尺度\(\ell_{\phi }\)的 DNS 案例进行了参数分析. 显式雷诺平均 DNS 数据表明，对于\(\widetilde{{\varepsilon_{c\xi } }}\)对于大多数情况来说是不够的，但它的性能似乎随着初始\(\ell_{\phi }\)和\(u^{\prime }\)值。\(\widetilde{{\varepsilon_{c\xi } }}\)的精确传输方程已经从第一原理推导出来，并且\(\widetilde{{\varepsilon_{c\ ) 的未闭合项的预算xi } }}\)输运方程进行了详细分析。已经发现由密度变化、标量-湍流相互作用、化学反应速率和分子耗散速率产生的项在\(\widetilde{{\varepsilon_{c\xi } }}\)运输。这些观察结果已通过缩放分析得到证明，该分析已被用于识别前导阶项的主要成分，以帮助\(\widetilde{{\varepsilon_{c\xi } }}\的未闭合项的模型开发)传输方程。新提出的未闭合项模型的性能已针对从 DNS 数据中提取的相应项进行了评估，并且新提出的闭包对\(\widetilde{{\varepsilon_{c\xi ) 中的未闭合项产生了令人满意的预测} }}\)传输方程。