Acoustic Destabilization of Vortices

E. G. Broadbent, D. W. Moore

Abstract

A line vortex which has uniform vorticity 2$\Omega _{0}$ in its core is subjected to a small two-dimensional disturbance whose dependence on polar angle is e$^{\text{i}m\theta}$. The stability is examined according to the equations of compressible, inviscid flow in a homentropic medium. The boundary condition at infinity is that of outgoing acoustic waves, and it is found that this capacity to radiate leads to a slow instability by comparison with the corresponding incompressible vortex which is stable. Numerical eigenvalues are computed as functions of the mode number m and the Mach number M based on the circumferential speed of the vortex. These are compared with an asymptotic analysis for the m = 2 mode at low Mach number in which it is found that the growth rate is ($\pi $/32)M$^{4}\Omega _{0}$ in good agreement with the numerical results.

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