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Algebraic instability caused by acoustic modes in supersonic shear flows

超音速シア流における音響モードが引き起こす代数的不安定性

廣田 真; 吉田 善章*

Hirota, Makoto; Yoshida, Zensho*

シア流中の揺らぎは複雑な挙動を示し、指数関数的な不安定モードが存在しない場合でも揺らぎは代数的な成長をする可能性がある。シア流は揺らぎの固有値問題を非エルミートにすると同時に、無限自由度系に特有な連続スペクトルをもたらす。本論文ではラプラス変換を用いて初期値問題を解くことにより、音響モード(点スペクトル)と渦の連続モード(連続スペクトル)の共鳴に起因する新しい代数的不安定性を見つけた。このような共鳴は速度シアの変化が音速に近くなると起こり得る。

Perturbations in a shear flow exhibit rather complex behavior - waves may grow algebraically even when the spectrum of disturbances is entirely neutral (no exponential instability). A shear flow brings about non-selfadjoint property, invalidating the standard notion of dispersion relations, and it also produces a continuous spectrum that is a characteristic entity in an infinite-dimension phase space. This paper solves an initial value problem using the Laplace transform and presents a new-type of algebraic instability that is caused by resonant interaction between acoustic modes (point spectrum) and vortical continuum mode (continuous spectrum). Such a resonance is possible when variation of velocity shear is comparable to sound speed.

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