In this work, we analyze the capacity of the superstatistics construction to provide modeling of the velocity field probability density functions (PDFs) of isotropic turbulence. Generalizing along the lines of the kappa distribution, superstatistics is understood here as a PDF for the statistical temperature that depends on a single dimensionful parameter ${\theta }^2$ and a dimensionless parameter ${\kappa }_0$, which both depend on the size of the fluid eddies and the Reynolds number, and possibly on auxiliary dimensionless constants that depend only on the Reynolds number. We show that such superstatistics –in some sense, the simplest class of models– cannot provide PDFs for scales outside the dissipation subrange for the currently accessible Reynolds numbers in Direct Numerical Simulations (DNS). The obstruction results from realizability constraints and an associated bound, and is related to the flatness factor of the velocity derivative distribution. Greater values of the flatness extend the applicability of superstatistics to larger scales. We argue that phenomenologically effective superstatistics models will require a value of flatness F$\sim$25 or larger in order to cover the inertial subrange scales. The argument is assisted by constructing and analyzing a family of models which derive from modifying the gamma distribution in the regime of large statistical temperatures and nearly realize the realizability bound.

在这项工作中，我们分析了超统计构造的能力，以提供各向同性湍流速度场概率密度函数（PDF）的建模。沿kappa分布线概括，超统计量在这里被理解为统计温度的PDF，其取决于单个维度参数${\theta }^2$ 和无量纲参数 ${\kappa }_0$，这两者都取决于流体涡流的大小和雷诺数，还可能取决于仅依赖于雷诺数的辅助无量纲常数。我们表明，从某种意义上说，这些超统计量-从某种意义上讲是最简单的模型-无法为直接数值模拟（DNS）中当前可访问的雷诺数的耗散子范围之外的尺度提供PDF。障碍源于可实现性约束和相关联的边界，并且与速度导数分布的平坦度因子有关。较高的平坦度值将超统计的适用范围扩展到了更大的规模。我们认为，在现象学上有效的超统计模型将需要平坦度F的值$〜$25或更大，以覆盖惯性子量程。通过构建和分析一系列模型来支持该论点，这些模型是通过在较大的统计温度范围内修改伽玛分布而得出的，并且几乎实现了可实现性范围。