Townsend's attached eddy hypothesis (AEH) gives an accurate phenomenological description of the flow kinematics in the logarithmic layer, but it suffers from two major weaknesses. First, AEH does not predict the constants in its velocity scalings, and second, none of the predicted velocity scalings can be obtained from the Navier-Stokes (NS) equations under AEH's assumptions. These two weaknesses separate AEH from more credible theories like Kolmogorov's theory of homogeneous isotropic turbulence, which, despite its phenomenological nature, has one velocity scaling, i.e., $〈\mathrm{\Delta }{u}^3〉=-(4/5)\epsilon r$, that can be derived from the NS equation. Here, $〈\mathrm{\Delta }{u}^3〉$ is the longitudinal third-order structure function, $\epsilon$ is the time-averaged dissipation rate, and $r$ is the displacement between the two measured points. This work aims to address these two weaknesses by investigating the behavior of the third-order structure function in the logarithmic layer of boundary-layer turbulence. We invoke AEH and obtain $〈\mathrm{\Delta }{u}^3〉=D_{3}ln(r/z)+B_3$, where $\mathrm{\Delta }u$ is the streamwise velocity difference between two points that are displaced by a distance $r$ in the streamwise direction, $z$ is the wall-normal location of the two points, $D_3$ is a universal constant, and $B_3$ is a constant. We then evaluate the terms in the Kármán-Howarth-Monin (KHM) equation according to AEH and see if NS equations give rise to a nontrivial result that is consistent with AEH. Last, by resorting to asymptotic matching, we determine $D_3=2.0$ (at sufficiently high Reynolds numbers).

中文翻译：

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边界层湍流对数层中的三阶结构函数

Townsend 的附加涡流假设 (AEH) 给出了对数层中流动运动学的准确现象学描述，但它有两个主要弱点。首先，AEH 不预测其速度标度中的常数，其次，在 AEH 的假设下，无法从 Navier-Stokes (NS) 方程中获得任何预测的速度标度。这两个弱点将 AEH 与更可信的理论区分开来，例如 Kolmogorov 的均匀各向同性湍流理论，尽管它具有现象学性质，但具有一个速度标度，即，$〈\mathrm{\Delta }{你}^3〉=-(4/5)\epsilon r$，这可以从 NS 方程推导出来。这里，$〈\mathrm{\Delta }{你}^3〉$ 是纵向三阶结构函数， $\epsilon$ 是时间平均耗散率，和 $r$是两个测量点之间的位移。这项工作旨在通过研究边界层湍流对数层中三阶结构函数的行为来解决这两个弱点。我们调用AEH并获得$〈\mathrm{\Delta }{你}^3〉=D_3输入(r/z)+{乙}_3$， 在哪里 $\mathrm{\Delta }你$ 是位移一定距离的两点之间的流向速度差 $r$ 在流线方向， $z$ 是两点的壁法线位置， $D_3$ 是一个普遍常数，并且 ${乙}_3$是一个常数。然后，我们根据 AEH 评估 Kármán-Howarth-Monin (KHM) 方程中的项，并查看 NS 方程是否产生与 AEH 一致的非平凡结果。最后，通过渐近匹配，我们确定$D_3=2.0$ （在足够高的雷诺数下）。