We study passive scalar fluctuations convected by statistically stationary homogeneous isotropic turbulence under a uniform mean scalar gradient. In order to elucidate the parameter dependence of small-scale statistics of scalar fluctuations, we conduct direct numerical simulations of passive scalar turbulence with 59 different combinations of Reynolds number and Schmidt number. For all the cases, we compute time-average statistics of various quantities, which include the scalar derivative skewness and flatness, the ratio of parallel-to-perpendicular scalar-gradient variances, and the anisotropy parameter recently proposed (Hill, Phys. Rev. Fluids, vol. 2, 2017, 094601). Notably, the degree of small-scale anisotropy of passive scalar fluctuation is characterised by a universal function of the Peclet number $Pe_{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}}=u^{\prime }\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D705}$ , where $u^{\prime }$ is the root mean square velocity, $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ the Taylor microscale of scalar fluctuation, $\unicode[STIX]{x1D705}$ the mass diffusivity. In the definition of the Peclet number, the use of $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ , rather than the Taylor microscale of velocity fluctuation, is key to collapsing the data of different Reynolds and Schmidt numbers. When the Peclet number is low, large-scale anisotropic scalar structures emerge irrespective of the Reynolds number. These structures are elongated along the direction of the uniform mean scalar gradient, and their size is significantly larger than the integral length scale of velocity fluctuation.

中文翻译：

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各向同性湍流均匀平均梯度下被动标量涨落的小尺度各向异性的 Péclet 数依赖性

我们研究了在均匀平均标量梯度下由统计平稳的均匀各向同性湍流对流产生的被动标量波动。为了阐明标量涨落的小尺度统计的参数依赖性，我们对具有 59 种不同雷诺数和施密特数组合的被动标量湍流进行了直接数值模拟。对于所有情况，我们计算各种数量的时间平均统计数据，其中包括标量导数偏度和平坦度、平行与垂直标量梯度方差的比率以及最近提出的各向异性参数（Hill, Phys. Rev.流体，第 2 卷，2017 年，094601）。尤其，被动标量涨落的小尺度各向异性程度的特征在于派克莱特数 $Pe_{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}}=u^{\prime }\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D705}$ ，其中 $u^{\prime }$ 是均方根速度，$\unicode [STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ 标量波动的泰勒微尺度，$\unicode[STIX]{x1D705}$ 质量扩散率。在 Peclet 数的定义中，使用 $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ 而不是速度波动的泰勒微尺度，是折叠不同数据的关键雷诺数和施密特数。当 Peclet 数较低时，无论雷诺数如何，都会出现大规模的各向异性标量结构。