Mach wave radiation of nonlinearly evolving supersonic instability modes in shear layers

Abstract

Shear flows at sufficiently high Mach numbers support instability waves which travel supersonically relative to the ambient free stream. Such supersonic modes are known to produce intense far-field sound in the form of Mach wave radiation. In this paper, the nonlinear evolution of supersonic modes and the associated Mach wave radiation are analysed in a self-consistent fashion by using the high-Reynolds-number matched asymptotic expansion in conjunction with the multiple-scale method. Attention is focused on the relatively weak disturbances for which the nonlinear effect is comparable with the non-parallel-flow effect. The nonlinear spatial development of the mode is described by an amplitude equation in which the nonlinear term is contributed by the critical layer. The directivity of the radiated Mach waves in the far field is determined explicitly in terms of the amplitude function. The analysis is applicable to plane mixing layers, boundary layers, and planar and circular jets. In particular, it is shown that for the last two flows, the radiated Mach waves are most intensive in a beam which is perpendicular to the Mach wave front and emanates from the streamwise position at which the instability mode attains its maximum amplitude. The theoretical results are compared with direct numerical simulation and experimental data, and favourable qualitative and quantitative agreement is obtained.