Instabilities in spatially periodic systems: application to secondary instabilities and complex surfaces
Abstract:Instabilities in spatially periodic flows are investigated. Two types of periodicity, corresponding to significantly different physical systems, can be distinguished. The first type of periodicity, the simplest, is geometric and results from the spatial repetition of a pattern or an obstacle that periodically deforms the flow. An example is the flow above a rough or superhydrophobic surface. The periodicity could also be generated by a modulation of the flow caused by the emergence of a coherent structure via a primary instability as in the case of cross-flow vortices. A framework allowing modal and non-modal stability analyses of spatially periodic systems is proposed. It is based on the Bloch theory and circulating matrices, and it is coupled with a two-dimensional stability problem. The framework is initially tested and validated on the secondary instability of cross-flow vortices. Stability of the flow over rough surfaces is then investigated, and notably the non-modal analysis displayed a parametric resonance between the roughness’ and the streaks’ wavelengths. Ultimately, the framework is also applied to the stability of the flow over a superhydrophobic surface. Effects of the interface deformation on the stability analysis are explicitly taken into account through a linearised Young-Laplace equation. A new family of unstable capillary modes is found and their role in the failure mechanisms of such surfaces is investigated.
