Uniform Asymptotic Approximations for Duct Acoustic Modes in a Thin Boundary-Layer Flow

Abstract

Analytical approximations for the acoustic modes in a duct carrying a uniform core flow with a thin shear layer at the walls are developed using the method of matched asymptotic expansions. Both two-dimensional and cylindrical duct propagation are considered. Numerical results for eigenvalues calculated using the theory are presented for the two-dimensional problem and compared with results from earlier analyses. It is found that the new approximations yield a significant increase in accuracy. HE problem of sound transmission through a duct carrying a uniform core flow with a thin boundary layer at the walls has been of interest for some time. Such a flow has practical applications in modeling a fully developed real flow profile. Perhaps the strongest motivation for studying the problem, however, is the possibility of obtaining analytical results using asymptotic analyses which can yield physical understanding of shear flow propagation phenomena in more realistic flows which must be treated numerically. Analytical studies based on the assumption of small boundary-layer thickness have been carried out by several researchers during the past decade. Eversman and Beckemeyer1 developed an inner expansion for acoustic pressure in the layer from which they were able to derive an approximate equivalent boundary condition applicable at the outer edge of the layer. Using this condition they showed that the formulation of the shear layer problem reduces to that of the associated uniform flow problem with the well-known particle displacement continuity boundary condition as the layer thickness vanishes. In a later paper, Eversman2 employed the equivalent boundary condition to formulate the eigenvalue problem for acoustic modes in an annular duct flow. Using the known uniform flow solutions in the core flow, an approximate eigenvalue equation is developed. Numerical solution of this single equation yields the acoustic wave numbers and circumvents the task of numerically integrating the differential equation and iterating for the shear flow eigenfunctions. The procedure is equally applicable to rectangular duct geometries, for which it leads to an eigenvalue equation equivalent to one later derived less formally by Tester. 3 A slightly different analysis by Swinbanks,4 also for rectangular ducts, yields similar approximate results. Other studies less directly related to the present work are discussed in Sec. IV.B ofRef. 5. The work reported in the present paper arose in the course of some research on shear flow propagation in which it was desirable to obtain analytical representations of the acoustic