Sound Propagation in Slowly Varying 2D Duct with Shear Flow

Sound propagation in uniformly lined straight ducts with uniform mean flow is well established by its analytically exact description in duct modes [1, 2, 3]. In cross-wise nonuniform flow there are still modes, although the (Pridmore-Brown) equation that describes them is in general not solvable in terms of standard functions [4, 5, 6, 7, 8, 9] and has to be solved numerically. These modal solutions provide insight, but the important effects due to the variation of the duct geometry and the corresponding mean flow cannot be described exactly. The inherently smooth variation of a flow duct, however, provides a small parameter (the slenderness of the duct wall variation) that allows asymptotic solutions of WKB type in the form of slowly varying modes in potential mean flow (asymptotically equivalent to quasi 1D gas flow in slender ducts) [10, 11, 12, 13, 14, 15, 16, 17, 19]. This approach have been favourably compared with fully numerical solutions [18], and has also otherwise been incorporated in full numerical approaches [17]. Much more difficult is it to combine slowly varying modes with non-uniform shear flow. The main problem is that a general theory for sheared flow in slowly varying ducts does not exist. An exception is the special case of uniform axial flow with solid body rotation, considered by Cooper and Peake in [20], but the mean flow still requires the numerical integration of a ordinary differential equation. If the duct varies just in impedance, rather than its geometry, like in [8], the problem of the mean flow is avoided and a rather general result is again possible. Another problem is that with nonuniform mean flow the acoustic equation (a form of the Pridmore-Brown equation) has in general no solutions in terms of standard functions, leading to a rather extended numerical component of the essentially analytical asymptotic approximation. In the present paper we will try to explore a special case of non-uniform shear flow in a slowly varying hard-walled duct, where both problems do not exist. As was found by Goldstein and Rice [21], the acoustic equations for 2D linear shear flow and uniform density, pressure and soundspeed, can be solved exactly by Weber’s Parabolic Cylinder functions, while the corresponding