On the acoustic modes in a cylindrical duct with an arbitrary wall impedance distribution

Abstract

The present paper considers the propagation of sound in a cylindrical duct, with a wall section of finite length covered by an acoustic liner whose impedance is an arbitrary function of position. The cases of (i) uniform wall impedance, and wall impedance varying along the (ii) circumference or (iii) axis of the duct, or (iv) both simultaneously, are explicitly considered. It is shown that a nonuniform wall impedance couples modes with distinct azimuthal l or axial m wave numbers, so that their radial wave numbers k can no longer be calculated separately for each pair (m,l). The radial wave numbers are the roots of an infinite determinant, in the case when the wall impedance varies either (i) circumferentially or (ii) radially. If the wall impedance varies (iv) both radially and circumferentially, then the radial wave numbers are the roots of a doubly infinite determinant, i.e., an infinite determinant in which each term is an infinite determinant. The infinite determinants specifying the radial wave numbers are written explicitly for sound in a cylindrical nozzle with a uniform axial flow, in which case the radial eigenfunctions are Bessel functions; the method of calculation of the radial wave numbers applies equally well to a cylindrical nozzle with shear flow and/or swirling flows, with the Bessel functions replaced by other eigenfunctions. The radial wave numbers are calculated by truncation of the infinite determinants, for several values of the aspect ratio, defined as the ratio of length to diameter. It is shown that a nonuniform wall impedance will give rise to additional modes compared with a uniform wall impedance. The radial wave numbers specify the eigenfrequencies for the acoustic modes in the duct; the imaginary parts of the eigenfrequencies specify the decay of the sound field with time, and thus the effectiveness of the acoustic liner.

Explicit approximation of the wavenumber for lined ducts

“…It has been shown that the accuracy can be improved by introducing empirical corrections and/or by using the explicit first iteration of the Newton method. These explicit approximations may be used easily for practical purposes: explicit one-mode determination can simplify models based on low frequency acoustic wave propagation and several applications, in 2D ducts [2][3][4][5][6][7] as well as Axisymmetric ducts [7][8][9][10][11], can be made simpler using this type of approximation at least as a starting point.…”

“…Since this dispersion relationship leads to a transcendental equation, there is no closed form expression for the wavenumber and iterative numerical methods are most often used. In view of the numerous applications of lined ducts [2][3][4][5][6][7][8][9][10][11], it could be very interesting to have an accurate explicit expression of the wavenumber rather than a numerical value.…”

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

“…The account on sound transmission from the aircraft noise sources to the interior of the near airport resident (Section 3) has included: (i) the spectral and directional broadening of noise [255][256][257][258][259][260][261]; (ii) the psychoacoustic distinctions between noise components. The noise mitigation measures (Section 4) mentioned include: (i) optimization of non-uniform acoustic liners [406][407][408][409][410][411][412][413][414][415][416] and use of partial chevron nozzle [417]; (ii) low noise operating procedures [432,433] consistent with flight safety and air traffic management rules [444][445][446][447][448][449][450][451][452][453][454]. Both can reduce noise exposure near airports [346][347][348][349][350][351][352][353][354][355][356][357]…”

“…The noise reduction provided by acoustic liners involves several penalties: (i) increased drag, hence loss of thrust, increased fuel consumption and emissions; (ii) the weight, volume, cost, installation and eventual maintenance or refurbishment of the liner. For a given weight or area of acoustic liner it is desirable to maximize the noise reduction by using [400][401][402][403][404][405][406][407][408][409][410][411][412][413][414][415][416] non-uniform liners tailored (Figure 12) to have higher impedance in regions of peak noise levels and lower impedance in regions of low noise. This tailoring of the non-uniform impedance of the liner to the sound field, may lead to adaptive liners, since the noise field of an engine changes with flight condition.…”

An improved multimodal method for sound propagation in nonuniform lined ducts

“…The axial wave numbers in the lined segment were then calculated by solving this set of equations using the method of Muller. Campos et al 10 studied acoustic modes in a cylindrical duct with an arbitrary wall impedance distribution with flow. When the wall impedance varies along the circumference, they calculate the acoustic modes in a similar way as Fuller.…”