The energy cascade is the most significant feature that separates turbulence from other unsteady flows, and results from the behavior of the nonlinear term in the Navier-Stokes equations. The mathematical form of this term, however, places constraints on exactly how it can act. Here we consider the action of the nonlinear term in physical space rather than in Fourier space, where the energy transfer between scales can be interpreted as a mechanical process where some scales do work on others. This formulation reveals the fundamental role played by geometry, as work can only be done when the eigenframes of the turbulent stress and strain rate are appropriately aligned. By comparing a direct numerical simulation of the Navier-Stokes equations, an ensemble of random solenoidal vector fields, and a random sampling of uniform eigenframe alignments, we show that this geometric alignment plays a much stronger role in determining the flux between scales than do the magnitudes of the stress and strain rate. We also show that when the alignment is effectively two dimensional, even when embedded in a three-dimensional flow, the energy flux is typically inverse, suggesting that the inverse cascade in two-dimensional turbulence may have a kinematic origin. Our results point to some potentially fruitful directions for turbulence modeling.

中文翻译：

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湍流叶栅中能量传递的几何约束

能量级联是将湍流与其他非恒定流分开的最重要特征，并且是由Navier-Stokes方程中非线性项的行为产生的。但是，该术语的数学形式对它的作用方式施加了约束。在这里，我们考虑非线性项在物理空间而非傅立叶空间中的作用，在傅立叶空间中，尺度之间的能量转移可以解释为机械过程，其中某些尺度在其他尺度上起作用。该公式揭示了几何学的基本作用，因为只有在湍流应力和应变率的本征框架适当对齐时才能完成工作。通过比较Navier-Stokes方程的直接数值模拟，随机螺线管矢量场的集合以及均匀特征帧排列的随机采样，我们表明，这种几何对齐方式在确定尺度之间的通量方面起着比应力和应变率的大小强得多的作用。我们还表明，当对准有效地为二维时，即使嵌入到三维流中，其能量通量也通常是相反的，这表明二维湍流中的反向级联可能具有运动学起源。我们的结果指出了湍流建模的一些可能富有成果的方向。这表明二维湍流中的逆级联可能是运动学起源的。我们的结果指出了湍流建模的一些可能富有成果的方向。这表明二维湍流中的逆级联可能是运动学起源的。我们的结果指出了湍流建模的一些可能富有成果的方向。