arXiv 2302.115642023-02-22physics.flu-dynphysics.comp-ph

超越最优扰动：瞬时增长的统计框架

Beyond optimal disturbances: a statistical framework for transient growth

Peter Frame, Aaron Towne

备注：: 32 pages, 19 figures

摘要：

瞬态增长理论描述了线性机制如何导致干扰的暂时放大，即使线性化系统是由其特征值定义的渐近稳定。这种增长传统上是量化的初始扰动，产生最大的响应，在能量增益方面，在其演变的峰值时间。虽然这限制了增长，但它可能大大夸大了真实扰动的增长。在本文中，我们介绍了一个统计的角度来看，瞬态增长模型的能量放大的扰动的统计。根据初始扰动的两点空间相关性，导出了平均能量放大的公式。我们还得到了一个准确的近似的概率密度函数的能量的增长扰动，从增长的置信界可以得到。将我们的分析应用于Poissonille流产生了一些观察结果。第一，平均增益可以大大小于最大值，特别是当干扰是宽带波数内容。在这些情况下，由于我们在概率密度函数中观察到的指数行为，几乎不可能实现接近最优的增长。其次，初始扰动的特征长度尺度对预期增长有显著影响;具体而言，大尺度初始扰动产生的数量级比小尺度的预期增长更大，这表明传入扰动的长度尺度可能是确定瞬时增长是否导致特定流过渡的关键。最后，虽然最佳的增长规模与雷诺数的平方，我们观察到，平均能量放大规模只有线性的初始相关性的某些合理的选择。

英文摘要：

The theory of transient growth describes how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified by finding the initial disturbance that generates the maximum response, in terms of energy gain, at the peak time of its evolution. While this bounds the growth, it can vastly overstate the growth of a real disturbance. In this paper, we introduce a statistical perspective on transient growth that models statistics of the energy amplification of the disturbances. We derive a formula for the mean energy amplification in terms of the two-point spatial correlation of the initial disturbance. We also derive an accurate approximation of the probability density function of the energy of the growing disturbance, from which confidence bounds on the growth can be obtained. Applying our analysis to Poisseuille flow yields a number of observations. First, the mean gain can be drastically smaller than the maximum, especially when the disturbances are broadband in wavenumber content. In these cases, it is exceedingly unlikely to achieve near-optimal growth due to the exponential behavior which we observe in the probability density function. Second, the characteristic length scale of the initial disturbances has a significant impact on the expected growth; specifically, large-scale initial disturbances produce orders-of-magnitude-larger expected growth than smaller scales, indicating that the length scale of incoming disturbances may be key in determining whether transient growth leads to transition for a particular flow. Finally, while the optimal growth scales quadratically with Reynolds number, we observe that the mean energy amplification scales only linearly for certain reasonable choices of the initial correlations.

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