Propagation of sound in ducts with shear flow

Abstract

Approximate calculations of flat duct wave guide modes based on the Ritz-Galerkin technique are presented. The advantages of the technique are in its ability to incorporate both wall shear layers (boundary layers), non-zero wall admittance characteristics, and to yield the mode functions so computed in the closed form of a family of simple polynomials. The shear layers render the medium dispersive even for the lowest mode on which they appear to have the maximum influence. The influence on all modes is accentuated as the frequency is increased or for upstream propagation. Thus the mode profiles computed are significantly different from those computed using uniform convection velocity. On the other hand, the cutoff frequencies do not appear to be appreciably altered, at least so long as the shear layers are thin. From the calculations presented it would seem that the continuous particle displacement boundary condition used with a slip velocity profile is an adequate approximation only at low frequencies and low Mach numbers. Even so, this approximate boundary condition (continuous particle displacement) performs better for downstream propagation compared to the upstream propagation. Otherwise, it becomes imperative, especially for the lower order modes at high frequencies, to include the more realistic no-slip velocity profile in the calculations.