Much of our theoretical understanding of statistically stable and unstable flows is from the classical Monin–Obukhov similarity theory: the theory predicts the scaling of the mean flow well, but its prediction of the turbulent fluctuation is far from satisfactory. This study builds on Monin–Obukhov similarity theory and Townsend’s attached-eddy hypothesis. We present a model that connects the mean flow and the streamwise velocity fluctuations in both neutral and unstable boundary-layer flows at both moderate and high Reynolds numbers. The model predictions are compared to direct numerical simulations of weakly unstable boundary layers at moderate Reynolds numbers, and large-eddy simulations of unstable boundary-layer flows at high Reynolds numbers. The flow is shear dominated. The range of stability parameter considered in this work is $L/\unicode[STIX]{x1D6FF}<-0.1$ , where $L$ is the Monin–Obukhov length, and $\unicode[STIX]{x1D6FF}$ is the boundary-layer height. Reasonably good prediction of velocity fluctuations based on knowledge of the mean velocity profile is obtained.

中文翻译：

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统计不稳定边界层流中速度波动的标度

我们对统计稳定和不稳定流动的大部分理论理解来自经典的 Monin-Obukhov 相似理论：该理论很好地预测了平均流量的比例，但其对湍流波动的预测远不能令人满意。本研究建立在 Monin-Obukhov 相似理论和 Townsend 的附加涡流假设的基础上。我们提出了一个模型，该模型将中等和高雷诺数下的中性和不稳定边界层流中的平均流和流向速度波动联系起来。将模型预测与中等雷诺数下弱不稳定边界层的直接数值模拟和高雷诺数下不稳定边界层流动的大涡模拟进行比较。流动以剪切为主。在这项工作中考虑的稳定性参数的范围是 $L/\unicode[STIX]{x1D6FF}<-0.1$ ，其中 $L$ 是 Monin–Obukhov 长度，$\unicode[STIX]{x1D6FF}$ 是边界层高度。获得了基于平均速度剖面知识的合理良好的速度波动预测。