Exact budget equations for the second-order structure function tensor $\langle \delta u_i \delta u_j \rangle$ are used to study the two-point statistics of velocity fluctuations in inhomogeneous turbulence. The Anisotropic Generalized Kolmogorov Equations (AGKE) describe the production, transport, redistribution and dissipation of every Reynolds stress component occurring simultaneously among different scales and in space, i.e. along directions of statistical inhomogeneity. The AGKE are effective to study the inter-component and multi-scale processes of turbulence. In contrast to more classic approaches, such as those based on the spectral decomposition of the velocity field, the AGKE provide a natural definition of scales in the inhomogeneous directions, and describe fluxes across such scales too. Compared to the Generalized Kolmogorov Equation, which is recovered as their half trace, the AGKE can describe inter-component energy transfers occurring via the pressure-strain term and contain also budget equations for the off-diagonal components of $\langle \delta u_i \delta u_j \rangle$. The non-trivial physical interpretation of the AGKE terms is demonstrated with three examples. First, the near-wall cycle of a turbulent channel flow at $Re_\tau=200$ is considered. The off-diagonal component $\langle -\delta u \delta v \rangle$, which can not be interpreted in terms of scale energy, is discussed in detail. Wall-normal scales in the outer turbulence cycle are then discussed by applying the AGKE to channel flows at $Re_\tau=500$ and $1000$. In a third example, the AGKE are computed for a separating and reattaching flow. The process of spanwise-vortex formation in the reverse boundary layer within the separation bubble is discussed for the first time.

中文翻译：

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非均匀湍流中的结构函数张量方程

二阶结构函数张量$\langle \delta u_i \delta u_j \rangle$的精确预算方程用于研究非均匀湍流中速度波动的两点统计。各向异性广义柯尔莫哥洛夫方程 (AGKE) 描述了在不同尺度和空间中（即沿统计不均匀性方向）同时发生的每个雷诺应力分量的产生、传输、再分配和耗散。AGKE 可有效研究湍流的组分间和多尺度过程。与更经典的方法（例如基于速度场谱分解的方法）相比，AGKE 提供了非均匀方向上的尺度的自然定义，并且也描述了跨这些尺度的通量。与广义柯尔莫哥洛夫方程相比，恢复为它们的半迹，AGKE 可以描述通过压力 - 应变项发生的组件间能量转移，并且还包含 $\langle \delta u_i \delta u_j \rangle$ 的非对角线分量的预算方程。AGKE 术语的非平凡物理解释通过三个示例进行演示。首先，考虑 $Re_\tau=200$ 处湍流通道流动的近壁循环。对非对角线分量 $\langle -\delta u \delta v \rangle$ 不能用尺度能量来解释，详细讨论。然后通过将 AGKE 应用于 $Re_\tau=500$ 和 $1000$ 的通道流来讨论外湍流循环中的壁面法向尺度。在第三个示例中，为分离和重新连接流计算 AGKE。