On Sound Emission by Acoustic Sources in a Shear Flow

Abstract

The Green's function for the plane unidirectional shear flow is obtained without restriction to either the high or low frequencies, that is in an exact form that applies also to local wavelengths comparable to the lengthscale of variation of the shear velocity of the mean flow. The method is a version of the variation of constants, and applies to any velocity profile for which the free wave solution of the unforced wave equation is available. The hyperbolic tangent shear velocity profile is taken as an example, and its eigenfunctions are used in the exact, closed expression for the sound field due to a monopole source in the shear flow. The acoustic pressure due to a point monopole source is plotted as a function of the coordinate transverse to the shear layer for several source positions in the shear flow and several values of the shear layer thickness, free stream Mach number and angle of incidence.

On the generalization to swirling flows of Lighthill’s eighth-power law. Part II: Laws of intensity of radiation for acoustic-vortical waves

“…The acoustic-vortical wave equation (61)≡(79b) for the pressure perturbation spectrum of a wave (57) of frequency ω and axial k and azimuthal m wavenumber specifies the radial dependence and involves the frequency with Doppler shift (79a) due to the nonuniform shear U and swirl Ω; the coefficients D , E , and source term Z~ are given by equations (62a,b; 63a,b). The acoustic-vortical wave equation (79b) in the absence of swirl Ω=0 reduces to the acoustic-shear wave equations (61; 62a,b; 63a,b) written most often for plane homentropic flow, 20–23 for which two sets of exact solutions are known: (i) for homentropic mean flow; 24–36 (ii) for nonhomentropic isentropic mean flow. 37,38 The wave equation (79b) has a singularity at critical layer where the Doppler shifted frequency (79a) vanishes (80a) …”

“…[39][40][41] Thus, a critical layer and a continuous spectrum exists for: (i) acoustic-vortical waves in any shear flow, e.g. plane homentropic with linear, [24][25][26][27][28][29][30][31][32][33][34][35][36][37] exponential, 32 hyperbolic tangent, 33,35 or parabolic 34 velocity profile; (ii) acoustic-vortical waves in any shear flow, e.g. plane homenergetic, that is isentropic but nonhomentropic; 37,38 (iii) acoustic-vortical waves in any swirling flow, with the exception of rigid body swirl.…”