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Variational necessary and sufficient stability conditions for inviscid shear flow.

M Hirota1, P J Morrison2, Y Hattori1

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Proceedings. Mathematical, Physical, and Engineering Sciences
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PubMed
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.

Keywords:
Kreĭn signaturecontinuum Hamiltonian Hopf bifurcationflow stabilityvariational method

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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.