A sensitivity-based protocol for identifying non-viscous mechanisms causing fluid dynamic instabilities

In recent decades, there has been a growing focus on investigating elementary vortex flows, largely fueled by a renewed interest in coherent structures, which play a fundamental role in many turbulent flows. Identifying critical conditions of these eddies is essential for understanding the flow behavior and the organization of fluid fields. While global stability analysis enables the identification of critical conditions, it does not offer insight into the mechanisms driving these instabilities. Lifschitz and colleagues [1], [2] have presented an alternative technique to traditional hydrodynamic instability theory known as the geometric optics approach. This methodology, which relies on the geometric properties of underlying base flows, offers a robust framework for establishing local instability criteria. The link between normal mode analysis and geometric optics investigation was initially explored by Sipp et al. [3], who delved into the stability of flattened Taylor-Green vortices. The researchers identified conditions leading to centrifugal, elliptic, and hyperbolic inviscid instabilities. However, pinpointing inviscid mechanisms in real flow configurations proves to be challenging, prompting the application of multiple methodologies (see e.g. [4]) to grasp the physical nature of the instabilities. In this study, we propose an effective approach to discern non-viscous mechanisms through structural sensitivity analysis. The key ingredient of our protocol lies in making viscosity effects vanish solely in the stability equations while maintaining the base flow field constant, computed at the critical Reynolds number. As a test case, we investigate the flow in a planar sudden expansion. Our findings indicate that classical structural sensitivity accurately identifies the instability core within the recirculation bubble. However, only the inviscid structural sensitivity field clearly indicate that the instability core is concentrated around the centre of an elliptical vortex: a clear indicator of elliptic instability. To validate our findings, we compare global sensitivity results with geometric optics outcomes.