×

zbMATH — the first resource for mathematics

Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes. (English) Zbl 0632.76074
Classical 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.

MSC:
76H05 Transonic flows
76Q05 Hydro- and aero-acoustics
76M99 Basic methods in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Artola and Majda, A. Nonlinear development of instabilities in supersonic vortex sheets. II. Resonant interaction among kink modes (to be submitted to Physica D). · Zbl 0692.76057
[2] M. Artola and A. Majda, Nonlinear development of acoustical instabilities in supersonic jets (to appear in Physica D). · Zbl 0653.76045
[3] M. Artola and A. Majda, Wave bifurcation and nonlinear travelling kink modes for supersonic vortex sheets (in preparation - to be submitted to Physics of Fluids).
[4] Begelman, M.C.; Blandford, R.D.; Rees, M.J., Theory of extragalactic radio sources, Rev. mod. physics, 56, 255-351, (1984)
[5] Cehelsky, P.; Rosales, R., Resonantly interacting weakly nonlinear hyperbolic waves in the presence of shocks: A single space variable in a homogeneous time-dependent medium, Stud. appl. math., 73, (1985)
[6] Courant, R.; Friedrichs, K., Supersonic flow and shock waves, () · Zbl 0365.76001
[7] DiPerna, R.; Majda, A., The validity of geometric optics for weak solutions of conservation laws, Comm. math. physics, 98, 313-347, (1985) · Zbl 0582.35081
[8] Higdon, R., Initial-boundary value problems for linear hyperbolic systems, SIAM review, 28, 177-217, (1986) · Zbl 0603.35061
[9] Hunter, J.K.; Keller, J.B., Weakly nonlinear high frequency waves, Comm. pure appl. math., 36, 547-569, (1983) · Zbl 0547.35070
[10] Hunter, J.K.; Majda, A.; Rosales, R., Resonantly interacting, weakly nonlinear hyperbolic waves II: several space variables, Stud. appl. math., 75, 187-226, (1986) · Zbl 0657.35084
[11] Lax, P.D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, () · Zbl 0108.28203
[12] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, () · Zbl 0537.76001
[13] Majda, A., Nonlinear geometric optics for hyperbolic systems of conservation laws, (), 116-166
[14] A. Majda and M. Artola, Nonlinear geometric optics for hyperbolic mixed problems, to appear in: Symposium in honor of Professor J.L. Lions’ sixtieth birthday.
[15] Majda, A.; Rosales, R., A theory for spontaneous Mach stem formation in reacting shock fronts, I: the basic perturbation analysis, SIAM J. appl. math., 43, 1310-1334, (1983) · Zbl 0544.76135
[16] Majda, A.; Rosales, R., A theory for spontaneous Mach stem formation in reacting shock fronts, II: steady wave bifurcations and the evidence for breakdown, Stud. appl. math., 71, 117-148, (1984) · Zbl 0584.76075
[17] Miles, J.W., On the reflection of sound at an interface of relative motion, J. acoust. soc. am., 29, 226-228, (1957)
[18] Miles, J.W., On the disturbed motion of a plane vortex sheet, J. fluid mech., 4, 538-552, (1958) · Zbl 0084.42002
[19] Smarr, L.L.; Norman, M.L.; Winkler, K.A., Shocks, interfaces and patterns in supersonic jets, Physica, 12D, 83-106, (1984)
[20] Woodward, P., Piecewise-parabolic methods for astrophysical fluid dynamics, (November 1983), L.L.L. preprint
[21] Woodward, P., Simulation of the Kelvin-Helmholtz instability of a supersonic slip surface with a piecewise parabolic method, (March 1984), L.L.L. preprint
[22] P. Woodward, Simulation of nonlinear instability in supersonic slip streams (unpublished calculations, L.L.L. and University of Minnesota 1985 and 1986) and personal communication.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.