The propagation of small-amplitude modes in an inviscid but sheared subsonic mean flow inside a duct is considered. For isentropic flow in a circular duct with zero swirl and constant mean flow density the pressure modes are described in terms of the eigenvalue problem for the Pridmore-Brown equation with Myers' locally reacting impedance boundary conditions. The key purpose of the paper is to extend the results of the numerical study of the spectrum for the case of lined ducts with uniform mean flow in Rienstra (Wave Motion, vol. 37, 2003b, p. 119), in order to examine the effects of the shear and wall lining. In the present paper this far more difficult situation is dealt with analytically. The high-frequency short-wavelength asymptotic solution of the problem based on the WKB method is derived for the acoustic part of the spectrum. Owing to the stiffness of the governing equations, an accurate numerical study of the spectral properties of the problem for mean flows with strong shear proves to be a non-trivial task which deserves separate consideration. The second objective of the paper is to gain theoretical insight into the properties of the hydrodynamic part of the spectrum. An analysis of hydrodynamic modes both in the short-wavelength limit and for the case of the narrow duct is presented. For simplicity, only the hard-wall flow configuration is considered.
Vilenski, G. G., & Rienstra, S. W. (2007). On hydrodynamic and acoustic modes in a ducted shear flow with wall lining. Journal of Fluid Mechanics, 583, 45-70. https://doi.org/10.1017/S0022112007006106