Statistical quantities in channel (Poiseuille), Couette, and pipe flow, including Reynolds stresses and their budgets, are studied for their dependence on the normalized distance from the wall $y^{+}$ and the friction Reynolds number ${\text{Re}}_{\tau }$. Any quantity $Q$ can be normalized in wall units, based on the friction velocity $u_{\tau }$ and viscosity $\nu$, and it is accepted that the physics of fully developed turbulence in ducts leads to standard results of the type $Q^{+}=f(y^{+},{\text{Re}}_{\tau })$, in which $f(...)$ means “only a function of,” and $f$ is different in different flow types. Good agreement between experiments and simulations is expected. We are interested in stronger properties for $Q$, generalized from those long recognized for the velocity, but not based on first principles. These include the law of the wall $Q^{+}=f\left(y^{+}\right)$; the logarithmic law for velocity; and for the Reynolds-number dependence, the possibility that at a given $y^{+}$ it is proportional to the inverse of ${\text{Re}}_{\tau }$, that is, $Q^{+}(y^{+},{\text{Re}}_{\tau })=f_{\infty }\left(y^{+}\right)+f_{\text{Re}}\left(y^{+}\right)/{\text{Re}}_{\tau }$. This has been proposed before, also on an empirical basis, and recent work by Luchini [Phys. Rev. Lett. 118, 224501 (2017)] for $Q\equiv U$ is of note. The question of whether the profiles are the same in all three flows, in other words, that there is a single function $f_{\infty }$, is still somewhat open. We arrive at different conclusions for different $Q$ quantities. The inverse-${\text{Re}}_{\tau }$ dependence is successful in some cases. Its failure for some of the Reynolds stresses can be interpreted physically by invoking “inactive motion,” following Townsend [The Structure of Turbulent Shear Flow, 2nd ed. (Cambridge University Press, Cambridge, 1976)] and Bradshaw [J. Fluid Mech. 30, 241 (1967)], but that is difficult to capture with any quantitative theory or turbulence model. The case of the boundary layer is studied, and it is argued that a direct generalization of ${\text{Re}}_{\tau }$ is questionable, which would prevent a sound extension from the internal flows.

中文翻译：

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从直接数值模拟推导的壁边界湍流的经验比例定律

研究了通道（Poiseuille），库埃特（Couette）和管道流量（包括雷诺应力及其预算）中的统计量是否依赖于与墙的归一化距离 ${ÿ}^{+}$ 和摩擦雷诺数 ${\text{关于}}_{\tau }$。任何数量$问$ 可以根据摩擦速度以墙为单位进行归一化 ${ü}_{\tau }$ 和粘度 $\nu$，并且公认的是，在管道中充分发展的湍流的物理特性导致该类型的标准结果 ${问}^{+}=F（{ÿ}^{+}，{\text{关于}}_{\tau }）$，其中 $F（...）$ 表示“仅是...的功能”，并且 $F$在不同的流量类型上是不同的。期望在实验和模拟之间达成良好的协议。我们对以下方面的更强特性感兴趣$问$，从对速度的长期认可中得出，但并非基于第一原理。这些包括隔离墙的法律${问}^{+}=F（{ÿ}^{+}）$; 速度的对数定律; 对于雷诺数依赖，在给定条件下的可能性${ÿ}^{+}$ 它与的倒数成比例 ${\text{关于}}_{\tau }$， 那是， ${问}^{+}（{ÿ}^{+}，{\text{关于}}_{\tau }）=F_{\infty }（{ÿ}^{+}）+F_{\text{关于}}（{ÿ}^{+}）/{\text{关于}}_{\tau }$。Luchini [ Phys。莱特牧师 118，224501（2017）]为$问\equiv ü$值得注意。在所有三个流程中配置文件是否相同的问题，换句话说，只有一个功能$F_{\infty }$，仍然有些开放。对于不同的结论，我们得出不同的结论$问$数量。反之${\text{关于}}_{\tau }$在某些情况下，依赖是成功的。遵循Townsend [湍流剪切流的结构，第二版]，可以通过调用“非活动运动”来从物理上解释其对于某些雷诺应力的破坏。（Cambridge University Press，Cambridge，1976）和Bradshaw [ J. Fluid Mech。 30，241（1967）]，但是这是难以用任何定量的理论或湍流模型捕获。对边界层的情况进行了研究，并认为对边界层的直接推广${\text{关于}}_{\tau }$ 这是有问题的，这将阻止内部流程的声音扩展。