A Frequency Selection Criterion in Spatially Developing Flows

TLDR

The study investigates whether spatially developing flows can support global modes or self‑excited linear resonances using the WKBJ approximation. The study shows that the dominant global mode can be deduced from the spatial variation of the local absolute frequency ω₀(X); specifically, systems without absolute instability cannot support temporally growing global modes with O(1) growth rates, while if the nearest singularity of ω₀(X) is a saddle point, the most unstable mode has a complex frequency ω₀(X_s) and grows only when ω₀(X_s) is positive.

Abstract

The possible existence of global modes or self‐excited linear resonances in spatially developing systems is explored within the framework of the WKBJ approximation. It is shown that the existence and properties of the dominant global mode may be deduced from the variations of the local absolute frequency ω 0 with distance X . The main results are summarized in two theorems: (1) A system with no region of absolute instability does not sustain temporally growing global modes with an O (1) growth rate. (2) If the singularity X , closest to the real X ‐axis of the complex function ω 0 ( X ) is a saddle point, the most unstable global mode has, to leading order in the WKBJ approximation, a complex frequency ω 0 ( X s ). Thus, it will be temporally growing only if ω 0 ( X s ) is positive.