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Variational necessary and sufficient stability conditions for inviscid shear flow
M Hirota1, P J Morrison2, Y Hattori1
1Institute of Fluid Science, Tohoku University , Sendai, Miyagi 980-8677, Japan.
Proceedings. Mathematical, Physical, and Engineering Sciences
|December 9, 2014
Summary
A new variational principle simplifies assessing the linear stability of inviscid parallel shear flows. This method links flow instability to positive eigenvalues of a self-adjoint operator, offering a more tractable approach than solving Rayleigh's equation directly.
Area of Science:
- Fluid dynamics
- Mathematical physics
- Stability theory
Background:
- Assessing the linear stability of inviscid parallel shear flows is crucial in fluid dynamics.
- Rayleigh's equation, a non-self-adjoint eigenvalue problem, is commonly used but can be computationally intensive.
- Existing methods may not be universally applicable to all stability problems.
Purpose of the Study:
- To formulate a necessary and sufficient condition for the linear stability of inviscid parallel shear flow.
- To develop a novel variational principle for stability analysis.
- To provide a more tractable method for determining flow stability.
Main Methods:
- Developed a novel variational principle for stability analysis.
- Assumed monotonic and analytic velocity profiles.
- Associated unstable eigenvalues of Rayleigh's equation with positive eigenvalues of a self-adjoint operator.
- Utilized concepts of Kreĭn signature for continuous spectra.
Main Results:
- Established a variational stability criterion based on maximizing a quadratic form.
- Demonstrated that stability determination is more tractable than direct solution of Rayleigh's equation.
- Showed the applicability to other stability problems in infinite-dimensional Hamiltonian systems.
Conclusions:
- The developed variational principle provides an effective and more tractable method for assessing linear stability of inviscid parallel shear flows.
- The criterion is theoretically and numerically advantageous compared to directly solving Rayleigh's equation.
- The approach has broader implications for stability problems in Hamiltonian systems.
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