重温对Kármán“常数”的探寻
The hunt for the Kármán "constant'' revisited
AI总结：
本文解决了压力驱动流动中指示函数受线性重叠项影响导致难以确定Kármán常数的问题，提出一种新方法同时确定$κ$和线性项系数，所得结果与中心线速度推导值一致。
摘要：
壁面对数律将近壁处的平均速度剖面（缩写为MVP）与外区连接起来，在壁面湍流中，它一百多年来一直是湍流研究的固定内容，但对于前因子的值（即Kármán“常数”$κ$的倒数）或其普适性，目前仍无普遍共识。定位MVP对数部分的常用诊断工具是寻找指示函数$Ξ$（等于壁面法向坐标$y^+$乘以平均速度导数$\dd U^+/\dd y^+$）为常数的区域。然而，在压力驱动流动（如槽道流和管流）中，$Ξ$显著受一个与壁面法向坐标成正比的项影响，该项在内展开中为$\mathcal{O}(\Reytau^{-1})$量级，但在重叠区向上移动到外展开时变为主要的$\mathcal{O}(1)$量级。这里我们表明，由于该线性重叠项的存在，需要远超$10^5$的$\Reytau$才能在槽道和管道中产生一个数量级的近似常数的$Ξ$。通过考虑内渐近展开（进行到$\mathcal{O}(\Reytau^{-1})$）与外展开主阶的共同部分，解决了该问题。该共同部分包含对数律与线性项$S_0 \,y^+\Reytau^{-1}$的叠加，对应于$Ξ$的线性部分；在槽道和管道中，该线性部分被内展开的各项掩盖至$y^+ \approx 500-1000$。设计了一种新且稳健的方法，可在当前可达到的$\Reytau$下同时确定压力驱动流动中的$κ$和$S_0$，所得$κ$与由中心线速度的雷诺数依赖性推导出的$κ$一致。与零压力梯度湍流边界层的比较进一步澄清了这些问题。
英文摘要：
The logarithmic law of the wall, joining the inner, near-wall mean velocity profile (abbreviated MVP) in wall-bounded turbulent flows to the outer region, has been a permanent fixture of turbulence research for over hundred years, but there is still no general agreement on the value of the pre-factor, the inverse of the Kármán ``constant'' $κ$, or on its universality. The choice diagnostic tool to locate logarithmic parts of the MVP is to look for regions where the indicator function $Ξ$ (equal to the wall-normal coordinate $y^+$ times the mean velocity derivative $\dd U^+/\dd y^+$) is constant. In pressure driven flows however, such as channel and pipe flows, $Ξ$ is significantly affected by a term proportional to the wall-normal coordinate, of order $\mathcal{O}(\Reytau^{-1})$ in the inner expansion, but moving up across the overlap to the leading $\mathcal{O}(1)$ in the outer expansion. Here we show that, due to this linear overlap term, $\Reytau$'s well beyond $10^5$ are required to produce one decade of near constant $Ξ$ in channels and pipes. The problem is resolved by considering the common part of the inner asymptotic expansion carried to $\mathcal{O}(\Reytau^{-1})$, and the leading order of the outer expansion. This common part contains a \textit{superposition} of the log law and a linear term $S_0 \,y^+\Reytau^{-1}$, and corresponds to the linear part of $Ξ$, which, in channel and pipe, is concealed up to $y^+ \approx 500-1000$ by terms of the inner expansion. A new and robust method is devised to simultaneously determine $κ$ and $S_0$ in pressure driven flows at currently accessible $\Reytau$'s, yielding $κ$'s which are consistent with the $κ$'s deduced from the Reynolds number dependence of centerline velocities. A comparison with the zero-pressure-gradient turbulent boundary layer further clarifies the issues.
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