A classification of duct modes based on surface waves

Abstract

For the relatively high frequencies relevant in a turbofan engine duct, the modes of a lined section may be classified in two categories: genuine acoustic 3D duct modes resulting from the finiteness of the duct geometry, and 2D surface waves that exist only near the wall surface in a way essentially independent of the rest of the duct. Per frequency and circumferential order there are at most four surface waves. They occur in two kinds: two acoustic surface waves that exist with and without mean flow, and two hydrodynamic surface waves that exist only with mean flow. The number and location of the surface waves depends on the wall impedance Z and mean flow Mach number. When Z is varied, an acoustic mode may change via small transition zones into a surface waves and vice versa.Compared to the acoustic modes, the surface waves behave-for example as a function of the wall impedance-rather differently as they have their own dynamics. They are therefore more difficult to find. A method is described to trace all modes by continuation in Z from the hard-wall values, by starting in an area of the complex Z-plane without surface waves.

Numerical study of acoustic modes in ducted shear flow

“…Hence, with the exception of the first few indices, n holds the relation R ζ 1 π n/κ 1 1. As a result, in agreement with the theoretical prediction (20), most of the axial wavenumber trajectories are close to the circles whose centres are shifted slightly to the left from their hard-wall values in the upper half-plane k (positive n) and to the right in the lower half-plane (negative n). As predicted by the theory, for the uniform mean flow and the mean velocity profile with thin near-wall shear-flow sublayers (a = 250) the radii of these circles slowly grow with n (i.e., away from the real axis).…”

“…If the shear flow region is relatively thick, as is the case with a = 50 in Figure 6c, the opposite tendency prevails and the circles which are further away from the real axis have smaller radii than in the zero mean flow case studied in Rienstra [20]. The results depicted in Figure 6c for a = 50, do not contradict the proposed theory, since the theory assumes that wall Mach number M d is non-zero.…”

“…The exception is the case of large |X | whereby two branches of acoustic surface modes can be identified. For the first time, these were found numerically in Rienstra [20] in the uniform mean flow circular duct case. For a ducted flow with shear the related main-order solution is…”

“…Here the "boundary-layer thickness" is defined as the distance from the wall to the radial position r = , where the local velocity is equal to 99% of the "free stream velocity" M max . As in Rienstra [20], maximum Mach number M max = 0.5, Helmholtz number ω = 5 and m = 1 are used throughout this section. The typical acoustic wavelength is thus 2π d/ ω ≈ 1.26d.…”

A comparison of impedance boundary conditions for flow acoustics

“…As expected the boundary conditions (4), (5) and (9) with s ¼ 1 yield the same results as the exact solution for normal incidence (y ¼ 901) for all cases. Eq.…”