We consider the bypass transition in a flat plate boundary layer subject to free-stream turbulence and compute the evolution of the second-order structure function of the streamwise velocity, $du^2(,) $, from the laminar to the fully turbulent region using DNS. In order to separate the contributions from laminar and turbulent events at the two points used to define $du(\vec{x},\vec{r})$, we apply conditional sampling based on the local instantaneous intermittency, $\tau$ (1 for turbulent and 0 for laminar events). Using $\tau (\vec{x},t)$, we define two-point intermittencies, ${\gamma }^{(TT)}$, ${\gamma }^{(LL)}$ and ${\gamma }^{(TL)}$ which physically represent the probabilities that both points are in turbulent or laminar patches, or one in turbulent and the other in a laminar patch, respectively. Similarly, we also define the conditionally-averaged structure functions, $⟨d{u}^2{⟩}^{(TT)}$, $⟨d{u}^2{⟩}^{(LL)}$ and $⟨d{u}^2{⟩}^{(TL)}$ and decompose $⟨d{u}^2⟩(\vec{x},\vec{r})$ in terms of these conditional averages. The derived expressions generalise existing decompositions of single-point statistics to two-point statistics. It is found that in the transition region, laminar streaky structures maintain their geometrical characteristics in the physical and scale space well inside the transition region, even after the initial break down to form turbulent spots. Analysis of the $⟨d{u}^2{⟩}^{(TT)}$ fields reveal that the outer mode is the dominant secondary instability mechanism. Further analysis reveals how turbulence spots penetrate the boundary layer and approach the wall. The peaks of $⟨d{u}^2{⟩}^{(TT)}$ in scale space appear in larger streamwise separations as transition progresses and this is explained by the strong growth of turbulent spots in this direction. On the other hand, the spanwise separation where the peak occurs remains relatively constant and is determined by the initial inception process. We also analyse the evolution of the two-point intermittency field, ${\gamma }^{(TT)}$, at different locations. In particular, we study the growth of the volume enclosed within an iso-surface of ${\gamma }^{(TT)}$ and notice that it increases in both directions, with the growth in the streamwise direction being especially large. The evolution of these conditional two-point statistics sheds light into the transition process from a different perspective and complements existing analyses using single-point statistics.

中文翻译：

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过渡边界层中条件平均二阶结构函数的演化

我们考虑了在平板边界层中绕过自由流湍流的旁路过渡，并计算了从层流到完全湍流区域的流向速度$ du ^ 2（，）$的二阶结构函数的演化使用DNS。为了将层流和湍流事件的贡献分开来定义两个点$dü（\vec{X}，\overrightarrow{\mathrm{[R}}）$，我们根据局部瞬时间歇性应用条件抽样， $\tau$（湍流为1，层流事件为0）。使用$\tau （\vec{X}，Ť）$，我们定义了两点间断 ${\gamma }^{（ŤŤ）}$， ${\gamma }^{（\mathrm{大号}\mathrm{大号}）}$ 和 ${\gamma }^{（Ť\mathrm{大号}）}$在物理上代表两个点分别在湍流或层状斑块中，或者一个在湍流中而另一个在层状斑块中的概率。同样，我们还定义了条件平均结构函数，$⟨d{ü}^2{⟩}^{（ŤŤ）}$， $⟨d{ü}^2{⟩}^{（\mathrm{大号}\mathrm{大号}）}$ 和 $⟨d{ü}^2{⟩}^{（Ť\mathrm{大号}）}$ 分解 $⟨d{ü}^2⟩（\vec{X}，\overrightarrow{\mathrm{[R}}）$根据这些条件平均值。派生的表达式将现有的单点统计分解为两点统计。发现在过渡区域，层状条纹结构在过渡区域内部的物理和尺度空间中也保持了其几何特征，即使在最初分解形成湍流点之后也是如此。分析$⟨d{ü}^2{⟩}^{（ŤŤ）}$场表明外部模式是主要的次要失稳机制。进一步的分析揭示了湍流点如何穿透边界层并接近壁。的高峰$⟨d{ü}^2{⟩}^{（ŤŤ）}$随着过渡的进行，尺度空间中的水流以较大的流向间隔出现，这可以通过在该方向上湍流点的强烈增长来解释。另一方面，出现峰值的沿翼展方向的间距保持相对恒定，并由初始起始过程确定。我们还分析了两点间断场的演变，${\gamma }^{（ŤŤ）}$，在不同的位置。特别是，我们研究了封闭在等值面内的体积的增长${\gamma }^{（ŤŤ）}$并注意到它在两个方向上都增加了，其中沿流向的增长特别大。这些有条件的两点统计的演变从不同的角度阐明了过渡过程，并补充了使用单点统计的现有分析。