On the Acoustics of Unbounded and Ducted Vortex Flows

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.

Section: Aeroacoustics Volume 12 • Number 7 and 8 • 2013mentioning

confidence: 94%

On Sound Emission by Acoustic Sources in a Shear Flow

Campos¹,

Kobayashi

2013

“…The uniform axial flow U=const and potential vortex swirl W~r -2 or V j~r -1 or constant circulation, is the only case [19][20][21] for which the mean flow is irrotational. For any other profile of angular velocity the mean flow is rotational leading to the acousticvortical wave equation [18][19][22][23][24] if the mean flow is homentropic, and to its extension if the mean flow is isentropic non-homentropic 5,18 .…”

Section: Acoustic Vortical Wave Equationmentioning

confidence: 99%

“…Instead a uniform axial flow is assumed here, and the critical layer is due to non-rigid body swirl. The special case of potential vortex swirl has been studied [19][20][21] . The other special case is rigid body rotation for which the acoustic-vortical wave equation has no critical layer.…”

Section: Uniform Axial Flow With Linear Swirlmentioning

confidence: 99%

“…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][20][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 .…”

Section: Introductionmentioning

confidence: 99%

“…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 exception of potential vortex swirl [19][20][21] , for which the angular velocity decays like the inverse square of the radius; the present paper considers the opposite case obtaining the analytical exact solution in the case of non-rigid body swirl with angular velocity proportional to the radius for an annular duct; the rigid or impedance wall boundary conditions specify the eigenvalues as the roots of a linear combination of the pressure and its radial derivative. Either case of rigid or impedance wall boundary conditions, leads to a linear system of…”

Section: Introductionmentioning

confidence: 99%

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On the Discrete and Continuous Spectrum of Acoustic-Vortical Waves

Campos¹,

Serrão²

2013

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 the generalization to swirling flows of Lighthill’s eighth-power law: Part I: Waves in axisymmetric sheared and swirling isentropic flows

Campos

2016

The Lighthill–Proudman theory of sound generation by turbulence, inhomogeneities, and dissipation in a medium at rest is extended to an axisymmetric mean flow with (i) arbitrary unidirectional shear velocity profile and (ii) arbitrary angular velocity of rotation, both depending only on the radius and also (iii) isentropic conditions. The forced acoustic-vortical wave equation is obtained in space-time and in frequency-wavenumber domain retaining the radial dependence. The wave operator describes the propagation of coupled acoustic-vortical waves including the special cases of potential vortex and rigid-body swirl. The forcing term specifies the compressive, shear, and swirl wave sources that model the generation of acoustic-vortical waves (part II).