A synthetic perturbative hypothesis for the multiscale analysis of the wake instability

The paper presents a nonparallel stability analysis of the intermediate region of the two-dimensional wake behind a bluff body. In particular, it analyzes the convective instabilities using a Wentzel-Kramers-Brillouin-Jeffreys method on a basic flow previously derived from intermediate asymptotics D. Tordella and M. Belan, Phys. Fluids 15, 1897, 2003. The multiscaling is carried out to explicitly account for the effects associated to the lateral momentum dynamics at a given Reynolds number. These effects are an important feature of the base flow and are included in the perturbative equation as well as in the associated modulation equation. At the first order in the multiscaling, the disturbance is locally tuned to the property of the instability, as can be seen in the zero-order theory near-parallel parametric Orr-Sommerfeld treatment. This leads to a synthetic analysis of the nonparallel correction of the instability characteristics. The system is, in fact, considered to be locally perturbed by waves with a wave number that varies along the intermediate wake and which is equal to the wave number of the dominant saddle point of the zero order dispersion relation, taken at different Reynolds numbers. In this study, the Reynolds number is thus the only parameter. It is shown that the corrections to the frequency, and to the temporal and spatial growth rates are remarkable in the first part of the intermediate wake and lead to absolute instability in regions that extend to about ten body scales. The correction increases with the Reynolds number and agrees with data from laboratory and numerical experiments in literature. An eigenfunction and eigenvalue asymptotic analysis for the far wake is included, which is in excellent agreement with the complete problem.