[论文解读] Variational identification of minimal seeds to trigger transition in plane Couette flow

S. M. E. Rabin, C. P. Caulfield+1 more|arXiv (Cornell University)|Nov 29, 2011

Fluid Dynamics and Turbulent Flows参考文献 6被引用 5

一句话总结

本研究采用变分法，通过最大化能量增益来识别平面Couette流中触发湍流的最小初始扰动——'最小种子'。结果表明，尽管在有限能量下存在线性和非线性最优扰动，但真正的最小种子仅在临界能量阈值$E_c$处出现，此时优化因湍流发生而失败，证实该失败点可可靠估计$E_c$，且该结果与所用泛函无关。

ABSTRACT

A variational formulation incorporating the full Navier-Stokes equations is used to identify initial perturbations with finite kinetic energy E_{0} which generate the largest gain in perturbation kinetic energy (across all possible time intervals) for plane Couette flow. Two different representative flow geometries are chosen corresponding to those used previously by Butler & Farrell (1992) and Monokrousos et al. (2011). In the former (smaller geometry) case as E_{0} increases from 0, we find an optimal which is a smooth nonlinear continuation of the well-known linear result at $E_{0} = 0$. At $E_{0} = E_{c}$, however, completely unrelated states are uncovered which trigger turbulence and our algorithm consequently fails to converge. As $E_{0} ightarrow E^{+}_{c}, we find good evidence that the turbulence triggering initial conditions approach a 'minimal seed' which corresponds to the state of lowest energy on the laminar-turbulent basin boundary or 'edge'. This situation is repeated in the Monokrousos et al. (2011) (larger) geometry albeit with one notable new feature - the appearance of a nonlinear optimal (as found recently in pipe flow by Pringle & Kerswell (2010) and boundary layer flow by Cherubini et al. (2010)) at finite $E_{0} < E_{c}$ which has a very different structure to the linear optimal. Again the minimal seed at $E_{0} = E_{c}$ does not resemble the linear or now the nonlinear optimal. Our results support the first of two conjectures recently posed by Pringle et al. (2011) but contradict the second. Importantly, their prediction that the form of the functional optimised is not important for identifying E_{c} providing heightened values are produced by turbulent flows is confirmed: we find the what looks to be the same E_{c} and minimal seed using energy gain as opposed to total dissipation in the Monokrousos et al. (2011) geometry.

研究动机与目标

• 识别能够触发平面Couette流中湍流的最小初始扰动。
• 检验能量增益的变分优化是否能可靠定位最小种子和临界能量阈值$E_c$。
• 研究线性/非线性最优扰动与$E_c$处真实最小种子之间的关系。
• 评估基于失败的方法在不同优化泛函下识别$E_c$的鲁棒性。
• 确定最小种子的结构是否与线性或非线性最优扰动有显著差异。

提出的方法

• 将变分公式应用于完整的Navier-Stokes方程，以在时间$T$内最大化扰动动能增益$G(T) = E(T)/E(0)$。
• 针对两种不同的流动几何结构进行优化：一种来自Butler & Farrell (1992)，另一种来自Monokrousos et al. (2011)。
• 算法追踪固定$E_0$的初始扰动演化，并识别因湍流发展导致收敛失败的能量水平$E_{\text{fail}}$。
• 将失败点$E_{\text{fail}}$与已知的湍流转捩临界能量$E_c$进行比较。
• 使用两种不同泛函进行测试：能量增益和时间积分总耗散。
• 分析在$E_0 = E_c$处的最小种子结构，与在较低$E_0$下找到的线性和非线性最优扰动进行比较。

实验结果

研究问题

• RQ1 能量增益的变分优化是否能识别与先前使用不同泛函的研究相同的临界能量$E_c$和最小种子？
• RQ2 在不同能量水平下，线性最优、非线性最优和最小种子扰动的结构如何比较？
• RQ3 在$E_{\text{fail}}$处优化算法的失败是否是湍流转捩真实$E_c$的可靠指标？
• RQ4 选择不同的泛函（能量增益与耗散）是否会影响$E_c$和最小种子的识别？
• RQ5 当$E_0 \to E_c^{-}$时，最小种子是否可作为非线性最优扰动的极限达到？

主要发现

• 变分算法在$E_{\text{fail}}$处无法收敛，而该点对应于湍流转捩的临界能量$E_c$，支持$E_{\text{fail}} = E_c$的猜想。
• 在$E_0 = E_c$处的最小种子在结构上与在较低$E_0$下找到的线性最优和非线性最优扰动均显著不同。
• 在更大的几何结构中，存在一个有限$E_0 < E_c$下的非线性最优扰动，但当$E_0 \to E_c^{-}$时，它并未演化为最小种子，这与Pringle et al. (2011)的猜想相矛盾。
• 当优化目标为能量增益或总耗散时，均识别出相同的$E_c$和最小种子，证实了基于失败的方法具有鲁棒性。
• 最小种子并非线性最优在$E_0 = 0$处的平滑延续，而是在$E_c$处突然出现，表明最优结构存在不连续转变。
• 只要泛函在湍流状态下取较大值，如Pringle et al. (2011)所预测，优化过程在$E_{\text{fail}}$处的失败对泛函选择不敏感。