Green's functions for a source in a mixing layer: direct waves, refracted arrival waves and instability waves

Abstract

Green's functions for a source embedded in an isothermal transversely sheared mixing layer are compared with direct numerical simulation (DNS) at various frequencies. Based on the third-order convective wave equation (Lilley 1974), three types of wave responses are analysed. For direct waves, a vortex sheet is used in the low-frequency limit, while in the high-frequency limit the procedure derived by Goldstein (1982) is re-visited. For refracted arrival waves propagating in the zone of silence, the vortex sheet model derived by Friedland & Pierce (1969) is re-visited in the low-frequency limit, while in the high-frequency limit the finite thickness model derived by Suzuki & Lele (2002) is applied. Instability waves excited by a very low-frequency source are formulated in the linear regime using the normal mode decomposition: eigen-functions are normalized using the adjoint convective wave equation, and the receptivity of instability waves is predicted. These theoretical predictions are compared with numerical simulations in two dimensions: DNS are performed based on the full Navier–Stokes equations (the free-stream Mach number is ). They also reveal that the receptivity is fairly sensitive to the Reynolds number and the source position within the mixing layer.