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General stability criterion of two-dimensional inviscid parallel flow

Liang Sun
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Abstract

General stability criterions of two-dimensional inviscid parallel flow are obtained analytically for the first time. First, a criterion for stability is found as $\frac{U''}{U-U_s}>-μ_1$ everywhere in the flow, where $U_s$ is the velocity at inflection point, $μ_1$ is eigenvalue of Poincaré's problem. Second, we also prove a principle that the flow is stable, if and only if all the disturbances with $c_r=U_s$ are neutrally stable. Finally, following this principle, a criterion for instability is found as $\frac{U''}{U-U_s}<-μ_1$ everywhere in the flow. These results extend the former theorems obtained by Rayleigh, Tollmien and Fjørtoft and will lead future works to investigate the mechanism of hydrodynamic instability.

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Liang Sun. 2005-12-27. General stability criterion of two-dimensional inviscid parallel flow. https://arxiv.org/abs/physics/0512208

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