Is the transition to unsteadiness in the wake of slender bodies an artefact of boundary conditions ?
David Fabre
TL;DR
The paper addresses the paradox of predicting transition to unsteadiness in slender-body wakes where no region of absolute instability is present. It demonstrates that arc-branch eigenmodes in global stability spectra arise from a nonlocal pressure feedback generated at the outlet of truncated domains, interacting with the convective wake and a trailing-edge receptivity mechanism. By deriving a nonlocal-pressure model and implementing downstream filtering methods (transverse sponge and complex mapping), the authors successfully suppress spurious arc-branch modes and recover physically meaningful onset predictions for the zero-incidence thin plate and the NACA0012 profile. The study then maps the instability thresholds for the NACA0012 wake at small incidences, highlighting how boundary conditions influence open-flow stability analyses and suggesting that similar nonlocal feedback mechanisms may operate in other strongly convective flows. These insights advance parametric stability analyses of open shear flows and prompt reconsideration of infinite-domain idealizations in practical applications.
Abstract
This work considers the transition to unsteadiness in the wake of 2D slender bodies, and questions the relevance of the generally accepted scenario involving a region of absolute instability within the near wake. The case of a thin plate at zero incidence is first considered. Despite the absence of absolute instability region, global stability analysis reveals the existence of numerous unstable eigenvalues organized along a characteristic "arc-branch" whose properties significantly depends upon the size of the numerical domain. These arc-branch modes are explained as resulting from a non-local pressure perturbation spuriously generated at the outlet of the domain due to the no-stress boundary condition, which then triggers the shedding of vortical structures at the trailing edge of the plate. The case of NACA0012 wing profiles at small incidences is then considered. Global stability analysis reveals that both the non-local spurious feedback mechanism and the classical local feedback mechanism are active. Trying to suppress the spurious feedback by enlarging the size of the numerical domain is shown to be inefficient. On the other hand, filtering methods suppressing the exponential spatial growth of perturbations, with either a sponge or a complex mapping, are found to be efficient. Thanks to these ideas, the critical Reynolds number and Strouhal number at onset can eventually be computed and are mapped for incidences in the range $ α\in [0^o - 5 ^o]$. It is postulated that the non-local feedback mechanism evidenced here could be at play in other strongly convective flows.