On the existence of spatially periodic temporally unstable disturbances in two-dimensional convectively unstable Prats flows

In linear and local stability analysis, the small amplitude disturbance behavior has been well understood for decades. In this context, convectively unstable flows act as noise amplifiers. In other words, a continuous excitation source at a fixed location triggers the spatial growth (or decay) of disturbances downstream of it. Hence, these flows have been traditionally analyzed using a spatial stability analysis, where disturbances are assumed temporally periodic and spatially unstable. Such a behavior has been consistently verified by direct numerical simulations. The present investigation shows that this agreement indeed occurs, but only close enough to the excitation source under certain parametric conditions. When the excitation source triggers disturbances that decay in space, far away from this source, spatially periodic and temporally unstable disturbances are allowed to grow and become dominant. Evidence for the scenario just described is provided using direct numerical simulations of the two-dimensional Prats problem.