Convective instability in wake intermediate asymptotics

Abstract

The paper presents a multiscale analysis of the intermediate region of the two-dimensional convectively unstable wake past a bluff body. A recent asymptotic expansion solution was used as basic flow (Tordella & Belan, Phys. Fluids, vol. 15, 2003, 1897). This solution was obtained by matching an inner to an outer flow, both of which are Navier–Stokes solutions. By introducing a spatio-temporal multiscaling into the instability problem, an inhomogeneous Orr–Sommerfeld equation and an associated modulation equation are obtained. The streamwise variation of the instability characteristics can then be deduced from the wave modulation, by considering the system to be perturbed by waves with a complex wavenumber that corresponds of the dominant saddle point of the local dispersion relation, taken at different positions downstream of the wake, and at different Reynolds numbers. The corrections of no parallelism are remarkable in the intermediate wake. When the disturbance is related to an early intermediate station, the corrections lead to absolute instability in the upstream portion of the intermediate wake, where, in addition, the spatial growth rate decreases. When the disturbance is related to a section in the far field, conditions of minimal temporal stability are reached about 20 body scales downstream. In the far field the temporal damping increases with the Reynolds number.