Collective secondary instabilities: an application to three-dimensional boundary-layer flow

• Published in: arXiv.org
• Main Authors: Jouin, Antoine, Cherubini, Stefania, Robinet, Jean-Christophe
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 02.08.2022
• Subjects: Bloch waves; Boundary layer flow; Coupling; Cross flow; Dimensional stability; Flow stability; Fluid flow; Stability analysis; Three dimensional flow; Two dimensional analysis; Wavelengths
• ISSN: 2331-8422

In some linearly unstable flows, secondary instability is found to have a much larger wavelength than that of the primary unstable modes, so that it cannot be recovered with a classical Floquet analysis. In this work, we apply a new formulation for capturing secondary instabilities coupling multiple length scales of the primary mode. This formulation, based on two-dimensional stability analysis coupled with a Bloch waves formalism originally described in Schmid et al. (2017), allows to consider high-dimensional systems resulting from several repetitions of a periodic unit, by solving an eigenproblem of much smaller size. Collective instabilities coupling multiple periodic units can be thus retrieved. The method is applied on the secondary stability of a swept boundary-layer flow subject to stationary cross-flow vortices, and compared with Floquet analysis. Two multi-modal instabilities are recovered: for streamwise wavenumber \(\alpha_v\) close to zero, approximately twelve sub-units are involved in large-wavelength oscillations; whereas a staggered pattern, characteristic of subharmonic instabilities, is observed for \(\alpha_v = 0.087\).