使用伴随方程追踪绝对/对流不稳定性之间的边界
Tracking the boundary between absolute/convective instability using adjoint equations
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中文总结 AI 辅助
研究如何确定绝对/对流不稳定性边界，开发直接延拓方法，将零群速度条件与多问题联立求解，通过缩放伪弧长延拓追踪解流形，恢复解析边界，相比传统方法更高效，能取代嵌套搜索和重建。
中文摘要 AI 辅助
传统上确定绝对/对流不稳定性边界需要在复波数平面中重复进行鞍点搜索，并随后扫描物理参数空间以定位零绝对增长。对于大型非正规特征值问题，这种嵌套计算成本高昂且对模态分支关联敏感。本文针对频率仿射广义特征值问题的中性定常鞍点边界开发了一种直接延拓方法。将零群速度条件表示为伴随可解性残差，并与直接和伴随特征值问题、复规范约束和中性增长条件一起求解。通过缩放伪弧长延拓追踪组合状态 - 参数空间中得到的一维解流形，无需切换物理延拓变量即可跨越参数折叠。该公式恢复了解析的金兹堡 - 朗道边界，对于高斯尾流奥尔 - 索末菲问题，在相对临界雷诺数上与单独制定的有限差分鞍点修正一致，约为\(10^{-8}\)。与嵌套的复波数和参数平面鞍点扫描相比，扫描计算所需的壁时间是直接伴随延拓的\(14.0\) - \(30.6\)倍，且随着重建边界的细化成本增加。应用于耦合的奥尔德罗伊德 - B自由表面膜揭示了中性鞍点流形的真正折叠以及所选鞍点族的重入CI - AI - CI边界几何形状。结果表明，伴随增强的伪弧长延拓可以通过直接且高效的计算追踪中性边界本身来取代嵌套鞍点搜索和参数平面重建。
英文摘要
Determining absolute/convective instability boundaries conventionally requires repeated saddle searches in the complex-wavenumber plane and a subsequent scan of the physical parameter space to locate zero absolute growth. Such nested calculations become costly and sensitive to modal branch association for large non-normal eigenvalue problems. This work develops a direct continuation method for neutral stationary-saddle boundaries of frequency-affine generalised eigenvalue problems. The zero-group-velocity condition is expressed as an adjoint solvability residual and solved together with the direct and adjoint eigenproblems, complex gauge constraints and the neutral-growth condition. The resulting one-dimensional solution manifold in the combined state--parameter space is tracked by scaled pseudo-arclength continuation, allowing parameter folds to be crossed without switching the physical continuation variable. The formulation recovers the analytical Ginzburg--Landau boundary and, for a Gaussian-wake Orr--Sommerfeld problem, agrees with separately formulated finite-difference saddle corrections to approximately $10^{-8}$ in relative critical Reynolds number. Compared with nested complex-wavenumber and parameter-plane saddle scanning, the tested scans require $8.1$--$52.2$ times the wall time of the direct adjoint continuation. Extrapolation of the measured cost--accuracy trend to a boundary error of $E_H\sim10^{-6}$ suggests an estimated cost ratio of approximately $1.8\times10^{4}$ in favour of the direct continuation. Application to a coupled Oldroyd--B free-surface film reveals genuine folds of the neutral-saddle manifold and a re-entrant CI--AI--CI boundary geometry for the selected saddle family.
