ABSTRACT Geometric stability theory is developed as an analogue of structural stability in physical space. Two steady flows are said to be geometrically equivalent if they have the same streamline pattern with velocities satisfying . A stationary solution to the Euler equations is non-unique if there exists a geometrically equivalent solution with horizontally varying F, in which case its geometric structure is stable and permits non-proportional velocity change. An Euler flow without such an equivalent solution is unique and geometrically unstable. Analysis of pseudo-plane flows shows that only constant-speed flows, specifically straightline jet and vertical-aligned circular vortex, are geometrically stable. The only stable flow with closed streamlines is vertical-aligned circular vortex, which provides a stability explanation for the phenomenon of vortex alignment and axisymmetrisation. A series of polynomial and nonpolynomial Euler solutions is used to validate the generic instability of pseudo-plane ideal flows. The quadratic solutions indicate that the pressure field has dynamic multiplicity and cannot be used as a proxy for geometric analysis.

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平稳欧拉流的几何稳定性

摘要 几何稳定性理论是作为物理空间中结构稳定性的类似物而发展起来的。如果两个稳定流具有相同的流线型且速度满足 ，则称它们在几何上是等价的。如果存在具有水平变化的 F 的几何等效解，则欧拉方程的平稳解是非唯一的，在这种情况下，其几何结构是稳定的并且允许非成比例的速度变化。没有这种等效解的欧拉流是唯一的且几何不稳定。伪平面流动的分析表明，只有恒速流动，特别是直线射流和垂直排列的圆形涡流，在几何上是稳定的。唯一具有封闭流线的稳定流动是垂直排列的圆形涡流，这为涡流排列和轴对称现象提供了稳定性解释。一系列多项式和非多项式欧拉解用于验证伪平面理想流的一般不稳定性。二次解表明压力场具有动态多重性，不能用作几何分析的代理。