The impedance boundary condition for acoustics in swirling ducted flow

Abstract

The acoustics of a straight annular lined duct containing a swirling mean flow is considered. The classical Ingard–Myers impedance boundary condition is shown not to be correct for swirling flow. By considering behaviour within the thin boundary layers at the duct walls, the correct impedance boundary condition for an infinitely thin boundary layer with swirl is derived, which reduces to the Ingard–Myers condition when the swirl is set to zero. The correct boundary condition contains a spring-like term due to centrifugal acceleration at the walls, and consequently has a different sign at the inner (hub) and outer (tip) walls. Examples are given for mean flows relevant to the interstage region of aeroengines. Surface waves in swirling flows are also considered, and are shown to obey a more complicated dispersion relation than for non-swirling flows. The stability of the surface waves is also investigated, and as in the non-swirling case, one unstable surface wave per wall is found.

The modified Myers boundary condition for swirling flow

“…It was shown independently by Eversman & Beckemeyer (1972) and Tester (1973) that the Myers boundary condition was the correct limit in the case of an infinitely thin boundary layer with no swirl. However, it was shown in Masson et al (2017) that this was not the correct limit for an infinitely thin boundary layer in swirling flow.…”

“…For convenience, the four coupled governing equations in (2.13) are arbitrarily considered rather than the two coupled differential equations (2.15). The latter choice is closer to the method of Khamis & Brambley (2016) and is described in Masson (2018), Masson et al (2017). By using (2.13), the scaling…”

“…After performing analysis similar to Masson et al (2017), and matching to the (constant) leading-order outer solution (see appendix B for the details), the leadingorder solutions are…”

“…It is worth computing the boundary condition in some useful limits. Firstly, it is clear that in the limit of an infinitely thin boundary layer with δ = 0, the boundary condition from Masson et al (2017) is recovered, i.e. in the limit of δ → 0, (3.26) tends to (2.19) provided that the slight assumption of constant density in the boundary layer is assumed.…”

“…It is often assumed that the Myers boundary condition is the limit of an infinitely thin boundary layer, and it was shown independently by Eversman & Beckemeyer (1972) and Tester (1973) that this is the case with no swirling flow. However, it was shown in Masson et al (2017) that it is not the correct limit for an infinitely thin boundary layer for swirling flow, with an additional term relating to the presence of centrifugal forces. Throughout this paper this will be referred to as the corrected Myers boundary condition.…”