On the Discrete and Continuous Spectrum of Acoustic-Vortical Waves

Abstract

A wave propagating in a non-uniform flow can have a critical layer where it is absorbed, amplified, reflected, or converted to another mode, possibly exchanging energy with the mean flow. Two examples are the propagation of: (i) fan noise in the shear flow in the air inlet of a jet engine; (ii) turbine noise in the swirling flow in the jet exhaust. Both situations (i) and (ii) are included by considering wave propagation in an axisymmetric isentropic non-homentopic mean flow allowing for the simultaneous existence of shear, swirl, temperature and density gradients. This corresponds to coupled acoustic-vortical modes that have both continuous and discrete spectra. It is shown that a critical layer exists where the Doppler shifted frequency vanishes. A stability condition is obtained for the continuous spectrum for all frequencies, axial and azimuthal wavenumbers generalizing previous results from the homentropic to the isentropic non-isentropic case. The wave fields are calculated and plotted for a case of non-rigid body rotation, namely angular velocity of swirl proportional to the radius; this demonstrates the mode conversion between acoustic and vortical waves across the critical layer where the waves are absorbed because the pressure spectrum vanishes. case of acoustic-shear waves for four velocity profiles: (i) linear 8-12 ; (ii) exponential for a boundary layer 13 ; (iii) hyperbolic tangent for a shear layer 14 ; (iv) parabolic for a ducted flow 15 . The case of an homenergetic shear flow with a linear velocity profile, that is isentropic but non-homentropic and leads to acoustic-shear waves, and has been considered without 16 and with 17 sound sources. The wave equation in an axisymmetric rotating flow that is for acoustic-swirl waves 5,18 has two simplest cases: (i) rigid body rotation 18 ; (ii) potential vortex swirl 19-21 . The present paper concerns an axisymmetric mean flow with shear and swirl, that has been considered in the homentropic case, corresponding to acoustic-vortical waves 22,23 using a velocity potential 24 . The extension to acoustic-vortical waves in an isentropic non-homentropic mean flow is discussed in the present paper, based on the wave for the pressure (section 2.1). This provides an extension of the stability condition 22,23 for the continuous spectrum of acoustic-vortical waves in an axisymmetric sheared and swirling mean flow to the non-homentropic case (section 2.2) allowing for entropy as well as pressure, density and temperature gradients that are adiabatically related.The Doppler shifted frequency is constant for uniform axial flow and rigid body swirl, excluding the existence of a critical layer. A critical layer exists in all other cases, for which the Doppler shifted frequency can vanish, leading to a singularity of the wave equation in: (i) a sheared axial flow as in an engine air intake; (ii) a rotating flow with non-rigid body swirl; (iii) both together (Figure 1). There are few studies of non-rigid body swirl as in a jet exhaust downstream of a turbine, with the excep...

On Physical Aeroacoustics with Some Implications for Low-Noise Aircraft Design and Airport Operations

“…Downstream of a turbine the mean flow is swirling, leading to acoustic-swirl waves [205][206][207][208][209][210][211][212][213][214][215]. Both the acoustic-shear waves in a fan inlet duct, and the acoustic-swirl waves in a turbine exaust duct are acoustic-vortical waves [216][217][218][219][220][221][222][223][224][225][226][227][228] with vorticity due to shear or swirl or both.…”

“…leads to the material derivative (11b) that appears in the wave equation for acoustic-vortical waves with shear and swirl [207,210,214,215]. For a wave Equation (10a) with Equation (12a) azimuthal wavenumber m:…”

“…In the Figure 2 was considered the shear flow upstream of the fan since swirl due to suction is small. The Figure 3 concerns the flow downstram of the turbine [215] that is both sheared (acoustic-shear waves) and swirling (acoustic-swirl waves). The acoustic-vortical waves with axial wavenumber k allow downstream k > 0 or upstream k < 0 propagation and clockwise m < 0 or counterclockwise m > 0 azimuthal modes.…”

“…The acoustic-vortical waves also interact with wall acoustic liners (Figures 11 and 12). Thus a fundamental difference is: (i) sound waves in a duct have a discrete spectrum, that is the normal transverse modes of a duct due to boundary conditions; (ii) acoustic-vortical waves have in addition a continuous spectrum [212][213][214][215] due to the energy exchange of sound with mean flow vorticity. The use of acoustic liners provides a means to absorb sound in a duct; the presence of shear or swirling flow leads to acoustic-vortical waves that affect the sound interaction with the liner: (i) there is sound absorption by the liner; (ii) the shear flow near the wall may further attenuate sound; (iii) the swirling flow has largest azimuthal velocity near the wall so that the vortical effect is strongest; (iv) in addition an acoustic liner may have a bias flow out of the liner [229] that also affects sound absorption.…”

On Sound Emission by Acoustic Sources in a Shear Flow

“…In this case of waves in an unidirectional shear flow the position of the critical layer depends on frequency and longitudinal wave number, so that the 'transition conditions' correspond to a continuous spectrum. The existence of a continuous spectrum applies to acoustic-vortical waves not only in a shear flow but also in a swirling flow [64][65][66].…”