Detuned secondary instabilities in three-dimensional boundary-layer flow

In this work we apply a formulation for capturing detuned secondary instabilities. This formulation, based on two-dimensional stability analysis coupled with a Bloch wave formalism originally described by Schmidt et al. [Phys. Rev. Fluids 2, 113902 (2017)2469-990X10.1103/PhysRevFluids.2.113902], allows us to consider high-dimensional systems resulting from several repetitions of a spatially periodic unit by solving an eigenproblem of much smaller size. Secondary instabilities coupling multiple periodic units thus can be retrieved. The method is applied on the secondary stability of a swept boundary-layer flow subject to stationary cross-flow vortices. Two distinct detuned secondary instabilities are retrieved. The first one, obtained for a detuning factor ϵ=0.35 and reaching a maximum growth rate for streamwise wave number αv=0.087, was already found in the work of Fischer and Dallmann [Phys. Fluids A: Fluid Dyn. 3, 2378 (1991)0899-821310.1063/1.858218]. The second instability is obtained for streamwise independent modes and small detuning factor ϵ=0.08. The corresponding mode presents large-wavelength oscillations, arising from a characteristic beating phenomenon. The physical origin of these two secondary instabilities has been investigated by varying the amplitude of the primary disturbance: for the latter instability, reminiscent of a type III mode, the unstable branch continuously deforms as the amplitude is increased, whereas a change of topology of the spectrum is observed for the αv=0 mode.