Effect of mean-velocity profile shapes on sound transmission through two-dimensional ducts

Acoustic propagation through a lined two-dimensional duct is examined in order to assess the influence of the shape of the mean-velocity profile on the attenuation rate. Five mean-velocity profiles are considered: linear, parabolic, Pohlhausen, 1/7th power law with a linear sub-layer, and linear with slip at the wall. It is shown that when the attenuation rate is tabulated as a function of the boundary-layer thickness, as is usually done, substantially different results are obtained from the several mean profiles. However, when the displacement thickness is used, a considerable collapse is achieved in the attenuation curves that are obtained from the various profiles. For downstream propagation all profiles produce essentially the same results over a reasonable range of boundary-layer thickness. However, for many cases of upstream propagation, the results from the “turbulent” boundary-layer profiles differ, significantly from the results of the other profiles even when compared on the basis of displacement thickness. For these cases, it is shown that the attenuation is a function of the shape factor in addition to the displacement thickness. In all cases, in the limit as the boundary-layer thickness vanishes, the numerical results approach those obtained from a uniform profile with continuity of particle displacement at the wall.