Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes.

*(English)*Zbl 0632.76074Classical linearized stability analysis predicts (neutral) stability of supersonic vortex sheets for compressible flow with normalized Mach numbers, \(M>\sqrt{2}\), while recent detailed numerical simulations indicate the nonlinear development of instabilities for \(M>\sqrt{2}\) through the development and interaction of propagating kink modes in the slip-stream. These kink modes are discontinuities in the slip-stream bracketed by shock waves and rarefaction waves which grow self-similarly in time.

The apparent paradox is resolved by developing appropriate small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions which yield the response to a very small amplitude nonlinear planar sound wave incident on the vortex sheet. The analysis leads to three specific angles of incidence depending on \(M>\sqrt{2}\) where nonlinear resonance occurs. For these three special resonant angles of incidence the perturbation expansions automatically yield simplified equations. These equations involve an appropriate Hamilton-Jacobi equation for the perturbed vortex sheet location; the derivative of the solution of this Hamilton-Jacobi equation provides boundary data for two nonlinear Burgers transport equations for the sound wave emanating from the two sides of the vortex sheet. These equations are readily solved exactly and lead to the quantitative time-dependent nonlinear development of three different types of kink modes with a structure similar to that observed earlier.

The apparent paradox is resolved by developing appropriate small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions which yield the response to a very small amplitude nonlinear planar sound wave incident on the vortex sheet. The analysis leads to three specific angles of incidence depending on \(M>\sqrt{2}\) where nonlinear resonance occurs. For these three special resonant angles of incidence the perturbation expansions automatically yield simplified equations. These equations involve an appropriate Hamilton-Jacobi equation for the perturbed vortex sheet location; the derivative of the solution of this Hamilton-Jacobi equation provides boundary data for two nonlinear Burgers transport equations for the sound wave emanating from the two sides of the vortex sheet. These equations are readily solved exactly and lead to the quantitative time-dependent nonlinear development of three different types of kink modes with a structure similar to that observed earlier.

##### MSC:

76H05 | Transonic flows |

76Q05 | Hydro- and aero-acoustics |

76M99 | Basic methods in fluid mechanics |

76L05 | Shock waves and blast waves in fluid mechanics |

##### Keywords:

linearized stability analysis; stability of supersonic vortex sheets; kink modes in the slip-stream; shock waves; rarefaction waves; small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions; small amplitude nonlinear planar sound wave; perturbation expansions; Hamilton-Jacobi equation; nonlinear Burgers transport equations
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\textit{M. Artola} and \textit{A. J. Majda}, Physica D 28, 253--281 (1987; Zbl 0632.76074)

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##### References:

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