We investigate the local energy flux rate ${\mathrm{\Pi }}_{\ell }\left(\mathbf{x}\right)$ towards small scales in isotropic turbulent flows using direct numerical simulations and applying different low-pass filters. Two different filters are examined: A sharp Fourier filter and a Gaussian filter. The probability density function (PDF) of the local energy flux is calculated for the different filters and for different filtering scales. It is shown that the local energy flux is a largely fluctuating quantity taking both negative and positive values and this is more pronounced for the sharp filter. The variance, skewness, and kurtosis of these fluctuations are shown to increase as the filtering scale is decreased. Furthermore, we calculate the joint PDF of ${\mathrm{\Pi }}_{\ell }\left(\mathbf{x}\right)$ with the local filtered strain rate $S_{\ell }$ and the enstrophy ${\mathrm{\Omega }}_{\ell }$. The flux shows a good correlation with the strain, but not with the enstrophy. It is shown that its conditional mean value scales like ${\langle {\mathrm{\Pi }}_{\ell }\rangle }_S\propto {\ell }^2{S}_{\ell }^3$ in support of the Smagorinsky eddy viscosity model. Nonetheless, strong fluctuations exist around this value that also need to be modeled. We discuss the implications of our results for subgrid scale models and propose new modeling directions.

中文翻译：

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湍流的局部能量通量

我们调查当地的能量通量率 ${\mathrm{\Pi }}_{\ell }（\mathbf{X}）$使用直接数值模拟并应用不同的低通滤波器，向各向同性湍流中的小尺度运动。检查了两个不同的滤镜：一个尖锐的傅立叶滤镜和一个高斯滤镜。对于不同的滤波器和不同的滤波比例，计算局部能量通量的概率密度函数（PDF）。结果表明，局部能量通量是一个很大的波动量，同时具有负值和正值，这对于尖锐的滤波器而言更为明显。这些波动的方差，偏度和峰度显示为随着过滤规模的减小而增加。此外，我们计算了${\mathrm{\Pi }}_{\ell }（\mathbf{X}）$ 与局部过滤应变率 ${\mathrm{小号}}_{\ell }$ 和内旋 ${\mathrm{\Omega }}_{\ell }$。通量显示出与应变的良好相关性，但与回旋率没有良好的相关性。结果表明，其条件均值的尺度像${〈{\mathrm{\Pi }}_{\ell }〉}_{\mathrm{小号}}\propto {\ell }^2{\mathrm{小号}}_{\ell }^3$支持Smagorinsky涡流粘度模型。但是，围绕该值存在很大的波动，也需要对其进行建模。我们讨论了结果对亚网格规模模型的影响，并提出了新的建模方向。