Small-scale turbulence can be comprehensively described in terms of velocity gradients, which makes them an appealing starting point for low-dimensional modeling. Typical models consist of stochastic equations based on closures for nonlocal pressure and viscous contributions. The fidelity of the resulting models depends on the accuracy of the underlying modeling assumptions. Here, we discuss an alternative data-driven approach leveraging machine learning to derive a velocity gradient model which captures its statistics by construction. We use a normalizing flow to learn the velocity gradient probability density function (PDF) from direct numerical simulation (DNS) of incompressible turbulence. Then, by using the equation for the single-time PDF of the velocity gradient, we construct a deterministic, yet chaotic, dynamical system featuring the learned steady-state PDF by design. Finally, utilizing gauge terms for the velocity gradient single-time statistics, we optimize the time correlations as obtained from our model against the DNS data. As a result, the model time realizations resemble the time series from DNS statistically closely.

中文翻译：

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通过归一化流为湍流速度梯度量身定制的模型


小尺度湍流可以用速度梯度来全面描述，这使它们成为低维建模的一个有吸引力的起点。典型模型由基于非局部压力和粘性贡献的闭包的随机方程组成。结果模型的保真度取决于基础建模假设的准确性。在这里，我们讨论了另一种数据驱动的方法，该方法利用机器学习来推导出速度梯度模型，该模型通过构造捕获其统计数据。我们使用归一化流从不可压缩湍流的直接数值模拟 （DNS） 中学习速度梯度概率密度函数 （PDF）。然后，通过使用速度梯度的单次 PDF 方程，我们构建了一个确定性但混沌的动力学系统，其特点是通过设计学习的稳态 PDF。最后，利用速度梯度单次统计的规范项，我们优化了从模型中获得的时间相关性与 DNS 数据。因此，模型时间实现在统计上与 DNS 的时间序列非常相似。