ABSTRACT We develop a theory for the cascade mixing terms in a moment closure approach to binary active scalar mixing in variable-density turbulence. To address the variable-density complications we apply, as a principle and constraint, the conservation of the probability density function (PDF) through a Fokker–Planck equation with bounded sample space whose attractor is the beta PDF with skewness. Mixing is related to a single-point PDF as a realisability principle to provide mathematically rigorous expressions for the small scale statistics in terms of largescale moments. The problem of the unknown small-scale mixing is replaced with the determination of the drift and diffusion terms of a Fokker–Planck equation in a beta-PDF-convergent stochastic process. We find that realisability of a beta-convergent process requires the mixing time-scale ratio, taken as a constant in passive scalar mixing, to be a function of the mean mass fraction, mean fluid density, the Atwood number, the density-volume correlation and moments of the density field. We develop and compare the new model with direct numerical simulations data of non-stationary homogeneous variable-density turbulence.

中文翻译：

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变密度湍流中混合的矩闭合的 Fokker-Planck 方法

摘要 我们在变密度湍流中的二元主动标量混合的矩闭合方法中开发了级联混合项的理论。为了解决可变密度的复杂性，我们应用概率密度函数 (PDF) 的守恒作为原则和约束，通过 Fokker-Planck 方程与有界样本空间，其吸引子是具有偏度的 beta PDF。混合与作为可实现性原则的单点 PDF 相关，以根据大尺度矩为小尺度统计提供数学上严格的表达式。未知的小尺度混合问题被在 beta-PDF 收敛随机过程中确定 Fokker-Planck 方程的漂移和扩散项所取代。我们发现，β 收敛过程的可实现性需要混合时标比，在被动标量混合中作为常数，是平均质量分数、平均流体密度、阿特伍德数、密度-体积相关性的函数和密度场的矩。我们开发新模型并将其与非平稳均匀变密度湍流的直接数值模拟数据进行比较。