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A Frequency Selection Criterion in Spatially Developing Flows
Author(s) -
Chomaz JeanMarc,
Huerre Patrick,
Redekopp Larry G.
Publication year - 1991
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1991842119
Subject(s) - saddle point , singularity , instability , mathematical analysis , mathematics , mode (computer interface) , saddle , function (biology) , physics , geometry , mechanics , mathematical optimization , computer science , evolutionary biology , biology , operating system
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.