Shock-Wave–Vortex Interactions: Shock and Vortex Deformations, and Sound Production

Abstract: In this paper we study the interaction of a shock wave with a cylindrical vortex. The objective of the study is to characterize the shock and vortex deformations and the mechanism of sound generation. The approach relies on the solution of the two-dimensional Euler equations by means of a high order finite volume weighted-ENO scheme. In order to provide some guidance into the analysis we have also developed an acoustic analogy of the problem by formulating a wave equation for the pressure disturbance that is solved analytically by means of Green's functions. A systematic study has been conducted by investigating the effects of vortex intensity and shock strength. Specifically, we have determined the dependence of shock distortion and vortex compression, and its subsequent nutation upon shock and vortex strengths. The acoustic field generated through shock--vortex interactions has been found to evolve in three stages and to exhibit a three sound quadrupolar directivity. In the early stages of the interaction the sound generated due to shock distortion shows a dipolar character, which then changes to a quadrupolar one due to a restoring mechanism that acts during the interaction of the shock with the rear part of the vortex. In the third stage secondary sounds are formed, which also show a quadrupolar directivity. The acoustic analogy applied to weak shock--vortex interactions also confirms the numerically predicted sound generation mechanism.

[1]  Parviz Moin,et al.  The free compressible viscous vortex , 1991, Journal of Fluid Mechanics.

[2]  T. Gatski,et al.  Analysis and simulation of a turbulent, compressible starting vortex , 1999 .

[3]  H. S. Ribner,et al.  Cylindrical sound wave generated by shock-vortex interaction , 1985 .

[4]  Sergio Pirozzoli,et al.  Shock wave–thermal inhomogeneity interactions: Analysis and numerical simulations of sound generation , 2000 .

[5]  Kristine R. Meadows,et al.  Computational Study on the Interaction Between a Vortex and a Shock Wave , 1991 .

[6]  The shock-vortex interaction: The origins of the acoustic wave , 1997 .

[7]  T. M. Weeks,et al.  Interaction of a starting vortex as well as a vortex street with a traveling shock wave , 1965 .

[8]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[9]  Gordon Erlebacher,et al.  Nonlinear Strong Shock Interactions: A Shock-Fitted Approach , 1998 .

[10]  Mark H. Carpenter,et al.  Computational Considerations for the Simulation of Shock-Induced Sound , 1998, SIAM J. Sci. Comput..

[11]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[12]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[13]  Elaine S. Oran,et al.  The interaction of a shock with a vortex: Shock distortion and the production of acoustic waves , 1995 .

[14]  L. Kovasznay,et al.  Non-linear interactions in a viscous heat-conducting compressible gas , 1958, Journal of Fluid Mechanics.

[15]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[16]  Ann P. Dowling,et al.  Sound and Sources of Sound , 1983 .

[17]  T. Poinsot.  Boundary conditions for direct simulations of compressible viscous flows , 1992 .

[18]  Y. Hattori,et al.  Sound generation by shock–vortex interactions , 1999, Journal of Fluid Mechanics.

[19]  M. D. Salas,et al.  A numerical study of two-dimensional shock vortex interaction , 1981 .

[20]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[21]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .